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Theorem intwun 8611
Description: The intersection of a collection of weak universes is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
intwun  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )

Proof of Theorem intwun
Dummy variables  x  u  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 445 . . . . . 6  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  C_ WUni )
21sselda 3349 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  u  e. WUni )
3 wuntr 8581 . . . . 5  |-  ( u  e. WUni  ->  Tr  u )
42, 3syl 16 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  Tr  u )
54ralrimiva 2790 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  Tr  u
)
6 trint 4318 . . 3  |-  ( A. u  e.  A  Tr  u  ->  Tr  |^| A )
75, 6syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  Tr  |^| A )
82wun0 8594 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  (/)  e.  u
)
98ralrimiva 2790 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  (/)  e.  u
)
10 0ex 4340 . . . . 5  |-  (/)  e.  _V
1110elint2 4058 . . . 4  |-  ( (/)  e.  |^| A  <->  A. u  e.  A  (/)  e.  u
)
129, 11sylibr 205 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  (/)  e.  |^| A )
13 ne0i 3635 . . 3  |-  ( (/)  e.  |^| A  ->  |^| A  =/=  (/) )
1412, 13syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
152adantlr 697 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  u  e. WUni )
16 intss1 4066 . . . . . . . . . 10  |-  ( u  e.  A  ->  |^| A  C_  u )
1716adantl 454 . . . . . . . . 9  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  |^| A  C_  u )
1817sselda 3349 . . . . . . . 8  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  /\  x  e.  |^| A
)  ->  x  e.  u )
1918an32s 781 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  x  e.  u )
2015, 19wununi 8582 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  U. x  e.  u )
2120ralrimiva 2790 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  U. x  e.  u
)
22 vex 2960 . . . . . . 7  |-  x  e. 
_V
2322uniex 4706 . . . . . 6  |-  U. x  e.  _V
2423elint2 4058 . . . . 5  |-  ( U. x  e.  |^| A  <->  A. u  e.  A  U. x  e.  u )
2521, 24sylibr 205 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  U. x  e.  |^| A
)
2615, 19wunpw 8583 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  ~P x  e.  u )
2726ralrimiva 2790 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  ~P x  e.  u
)
2822pwex 4383 . . . . . 6  |-  ~P x  e.  _V
2928elint2 4058 . . . . 5  |-  ( ~P x  e.  |^| A  <->  A. u  e.  A  ~P x  e.  u )
3027, 29sylibr 205 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  ~P x  e.  |^| A
)
3115adantlr 697 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  u  e. WUni )
3219adantlr 697 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  x  e.  u )
3316adantl 454 . . . . . . . . . 10  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  |^| A  C_  u )
3433sselda 3349 . . . . . . . . 9  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  /\  y  e.  |^| A )  ->  y  e.  u )
3534an32s 781 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  y  e.  u )
3631, 32, 35wunpr 8585 . . . . . . 7  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  { x ,  y }  e.  u )
3736ralrimiva 2790 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  A. u  e.  A  { x ,  y }  e.  u )
38 prex 4407 . . . . . . 7  |-  { x ,  y }  e.  _V
3938elint2 4058 . . . . . 6  |-  ( { x ,  y }  e.  |^| A  <->  A. u  e.  A  { x ,  y }  e.  u )
4037, 39sylibr 205 . . . . 5  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  { x ,  y }  e.  |^| A )
4140ralrimiva 2790 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. y  e.  |^| A { x ,  y }  e.  |^| A
)
4225, 30, 413jca 1135 . . 3  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  -> 
( U. x  e. 
|^| A  /\  ~P x  e.  |^| A  /\  A. y  e.  |^| A { x ,  y }  e.  |^| A
) )
4342ralrimiva 2790 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. x  e.  |^| A ( U. x  e.  |^| A  /\  ~P x  e.  |^| A  /\  A. y  e.  |^| A { x ,  y }  e.  |^| A
) )
44 simpr 449 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  =/=  (/) )
45 intex 4357 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
4644, 45sylib 190 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
47 iswun 8580 . . 3  |-  ( |^| A  e.  _V  ->  (
|^| A  e. WUni  <->  ( Tr  |^| A  /\  |^| A  =/=  (/)  /\  A. x  e.  |^| A ( U. x  e.  |^| A  /\  ~P x  e.  |^| A  /\  A. y  e.  |^| A { x ,  y }  e.  |^| A
) ) ) )
4846, 47syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  ( |^| A  e. WUni  <->  ( Tr  |^| A  /\  |^| A  =/=  (/)  /\  A. x  e.  |^| A ( U. x  e.  |^| A  /\  ~P x  e. 
|^| A  /\  A. y  e.  |^| A {
x ,  y }  e.  |^| A ) ) ) )
497, 14, 43, 48mpbir3and 1138 1  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726    =/= wne 2600   A.wral 2706   _Vcvv 2957    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   {cpr 3816   U.cuni 4016   |^|cint 4051   Tr wtr 4303  WUnicwun 8576
This theorem is referenced by:  wunccl  8620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-pw 3802  df-sn 3821  df-pr 3822  df-uni 4017  df-int 4052  df-tr 4304  df-wun 8578
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