Theorem wunop 9544

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

Hypotheses

Ref	Expression
wun0.1	|- ( ph -> U e. WUni )
wunop.2	|- ( ph -> A e. U )
wunop.3	|- ( ph -> B e. U )

Assertion

Ref	Expression
wunop	|- ( ph -> <. A , B >. e. U )

Proof of Theorem wunop

Step	Hyp	Ref	Expression
1		wunop.2	. . 3 |- ( ph -> A e. U )
2		wunop.3	. . 3 |- ( ph -> B e. U )
3		dfopg 4400	. . 3 |- ( ( A e. U /\ B e. U ) -> <. A , B >. = { { A } , { A , B } } )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> <. A , B >. = { { A } , { A , B } } )
5		wun0.1	. . 3 |- ( ph -> U e. WUni )
6	5, 1	wunsn 9538	. . 3 |- ( ph -> { A } e. U )
7	5, 1, 2	wunpr 9531	. . 3 |- ( ph -> { A , B } e. U )
8	5, 6, 7	wunpr 9531	. 2 |- ( ph -> { { A } , { A , B } } e. U )
9	4, 8	eqeltrd 2701	1 |- ( ph -> <. A , B >. e. U )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177   {cpr 4179   <.cop 4183  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-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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wunot  9545  wunress  15940  1strwunbndx  15981  catcoppccl  16758  catcfuccl  16759  catcxpccl  16847