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Theorem wuntp 9137
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
StepHypRef Expression
1 tpass 4095 . 2  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )
2 wununi.1 . . 3  |-  ( ph  ->  U  e. WUni )
3 dfsn2 4009 . . . 4  |-  { A }  =  { A ,  A }
4 wununi.2 . . . . 5  |-  ( ph  ->  A  e.  U )
52, 4, 4wunpr 9135 . . . 4  |-  ( ph  ->  { A ,  A }  e.  U )
63, 5syl5eqel 2514 . . 3  |-  ( ph  ->  { A }  e.  U )
7 wunpr.3 . . . 4  |-  ( ph  ->  B  e.  U )
8 wuntp.3 . . . 4  |-  ( ph  ->  C  e.  U )
92, 7, 8wunpr 9135 . . 3  |-  ( ph  ->  { B ,  C }  e.  U )
102, 6, 9wunun 9136 . 2  |-  ( ph  ->  ( { A }  u.  { B ,  C } )  e.  U
)
111, 10syl5eqel 2514 1  |-  ( ph  ->  { A ,  B ,  C }  e.  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1868    u. cun 3434   {csn 3996   {cpr 3998   {ctp 4000  WUnicwun 9126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-v 3083  df-un 3441  df-in 3443  df-ss 3450  df-sn 3997  df-pr 3999  df-tp 4001  df-uni 4217  df-tr 4516  df-wun 9128
This theorem is referenced by:  catcfuccl  15992  catcxpccl  16080
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