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Theorem wunpr 9141
Description: A weak universe is closed under pairing. (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 )
Assertion
Ref Expression
wunpr  |-  ( ph  ->  { A ,  B }  e.  U )

Proof of Theorem wunpr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wununi.2 . 2  |-  ( ph  ->  A  e.  U )
2 wunpr.3 . 2  |-  ( ph  ->  B  e.  U )
3 wununi.1 . . 3  |-  ( ph  ->  U  e. WUni )
4 iswun 9136 . . . . 5  |-  ( U  e. WUni  ->  ( U  e. WUni  <->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
54ibi 244 . . . 4  |-  ( U  e. WUni  ->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) )
65simp3d 1019 . . 3  |-  ( U  e. WUni  ->  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) )
7 simp3 1007 . . . 4  |-  ( ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  A. y  e.  U  { x ,  y }  e.  U )
87ralimi 2815 . . 3  |-  ( A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U )  ->  A. x  e.  U  A. y  e.  U  { x ,  y }  e.  U )
93, 6, 83syl 18 . 2  |-  ( ph  ->  A. x  e.  U  A. y  e.  U  { x ,  y }  e.  U )
10 preq1 4079 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
1110eleq1d 2491 . . 3  |-  ( x  =  A  ->  ( { x ,  y }  e.  U  <->  { A ,  y }  e.  U ) )
12 preq2 4080 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
1312eleq1d 2491 . . 3  |-  ( y  =  B  ->  ( { A ,  y }  e.  U  <->  { A ,  B }  e.  U
) )
1411, 13rspc2va 3192 . 2  |-  ( ( ( A  e.  U  /\  B  e.  U
)  /\  A. x  e.  U  A. y  e.  U  { x ,  y }  e.  U )  ->  { A ,  B }  e.  U
)
151, 2, 9, 14syl21anc 1263 1  |-  ( ph  ->  { A ,  B }  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   (/)c0 3761   ~Pcpw 3981   {cpr 4000   U.cuni 4219   Tr wtr 4518  WUnicwun 9132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-un 3441  df-in 3443  df-ss 3450  df-sn 3999  df-pr 4001  df-uni 4220  df-tr 4519  df-wun 9134
This theorem is referenced by:  wunun  9142  wuntp  9143  wunsn  9148  wunop  9154  intwun  9167  wuncval2  9179
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