Theorem wunot 9545

Description: A weak universe is closed under ordered triples. (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 )
wunot.3	|- ( ph -> C e. U )

Assertion

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

Proof of Theorem wunot

Step	Hyp	Ref	Expression
1		df-ot 4186	. 2 |- <. A , B , C >. = <. <. A , B >. , C >.
2		wun0.1	. . 3 |- ( ph -> U e. WUni )
3		wunop.2	. . . 4 |- ( ph -> A e. U )
4		wunop.3	. . . 4 |- ( ph -> B e. U )
5	2, 3, 4	wunop 9544	. . 3 |- ( ph -> <. A , B >. e. U )
6		wunot.3	. . 3 |- ( ph -> C e. U )
7	2, 5, 6	wunop 9544	. 2 |- ( ph -> <. <. A , B >. , C >. e. U )
8	1, 7	syl5eqel 2705	1 |- ( ph -> <. A , B , C >. e. U )

Syntax hints:   -> wi 4   e. wcel 1990   <.cop 4183   <.cotp 4185  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-ot 4186  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by: (None)