Theorem wuntp 9533

Description: A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	|- ( ph -> U e. WUni )
wununi.2	|- ( ph -> A e. U )
wunpr.3	|- ( ph -> B e. U )
wuntp.3	|- ( ph -> C e. U )

Assertion

Ref	Expression
wuntp	|- ( ph -> { A , B , C } e. U )

Proof of Theorem wuntp

Step	Hyp	Ref	Expression
1		tpass 4287	. 2 |- { A , B , C } = ( { A } u. { B , C } )
2		wununi.1	. . 3 |- ( ph -> U e. WUni )
3		dfsn2 4190	. . . 4 |- { A } = { A , A }
4		wununi.2	. . . . 5 |- ( ph -> A e. U )
5	2, 4, 4	wunpr 9531	. . . 4 |- ( ph -> { A , A } e. U )
6	3, 5	syl5eqel 2705	. . 3 |- ( ph -> { A } e. U )
7		wunpr.3	. . . 4 |- ( ph -> B e. U )
8		wuntp.3	. . . 4 |- ( ph -> C e. U )
9	2, 7, 8	wunpr 9531	. . 3 |- ( ph -> { B , C } e. U )
10	2, 6, 9	wunun 9532	. 2 |- ( ph -> ( { A } u. { B , C } ) e. U )
11	1, 10	syl5eqel 2705	1 |- ( ph -> { A , B , C } e. U )

Syntax hints:   -> wi 4   e. wcel 1990   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180  df-tp 4182  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  catcfuccl  16759  catcxpccl  16847