Selecting random distinct and disjoint lists in a larger list

I'm coding in Sage 9.1 (Or Python 3).

I have a fairly large list big_list, each element of big_list is itself a list of constant small length t. For example for t=8, there are about 40.000 lists in big_list.

I want to randomly select in big_list, n lists (n is fixed, between 1 and t-1), such that the n elements are all distinct and all pairwise disjoint : For example if [a,c] and [a,d] are two element of big_list I cannot select them together. I know that such a selection always exist given some constraint I have on t and n

The way I do so far : I start by selecting one random list random_list in big_list, then using a list comprehension to create a new big_list containing only the elements not intersecting random_list. The intersection is checked by the implicit booleanness of the empty list. However this part of my code is by far the slowest in term of runtime (it is used many many times), any improvement would be great.

Here is a minimal working example. Any help is appreciated, thanks.

import random

def intersection(list1, list2):
    """Intersection of two lists"""
    temp = set(list2)
    list3 = [value for value in list1 if value in temp]
    return list3

t=4
n=2

output=list()

#example of big_list, here I could select elements 0 and 2, or elements 1 and 3
#note that for any first choice of random_list, it is always possible to select a second one.
big_list=[[1,2,3,4],[1,2,5,6],[5,6,7,8],[3,4,7,8]]

random_list = random.choice(big_list)
output.append(random_list) 

#looping on the number of disjoint lists I need
for i in range(1,n):
    #using a list comprehension to create a new big_list containing only the elements not intersecting random_list
    #the intersection is checked by the implicit booleanness of the empty list
    big_list = [l for l in big_list if not intersection(l,random_list)]

    random_list = random.choice(big_list)
    output.append(random_list) 
    
output