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Theorem intwun 8357
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 443 . . . . . 6  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  C_ WUni )
21sselda 3180 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  u  e. WUni )
3 wuntr 8327 . . . . 5  |-  ( u  e. WUni  ->  Tr  u )
42, 3syl 15 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  Tr  u )
54ralrimiva 2626 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  Tr  u
)
6 trint 4128 . . 3  |-  ( A. u  e.  A  Tr  u  ->  Tr  |^| A )
75, 6syl 15 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  Tr  |^| A )
82wun0 8340 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  (/)  e.  u
)
98ralrimiva 2626 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  (/)  e.  u
)
10 0ex 4150 . . . . 5  |-  (/)  e.  _V
1110elint2 3869 . . . 4  |-  ( (/)  e.  |^| A  <->  A. u  e.  A  (/)  e.  u
)
129, 11sylibr 203 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  (/)  e.  |^| A )
13 ne0i 3461 . . 3  |-  ( (/)  e.  |^| A  ->  |^| A  =/=  (/) )
1412, 13syl 15 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
152adantlr 695 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  u  e. WUni )
16 intss1 3877 . . . . . . . . . 10  |-  ( u  e.  A  ->  |^| A  C_  u )
1716adantl 452 . . . . . . . . 9  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  |^| A  C_  u )
1817sselda 3180 . . . . . . . 8  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  /\  x  e.  |^| A
)  ->  x  e.  u )
1918an32s 779 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  x  e.  u )
2015, 19wununi 8328 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  U. x  e.  u )
2120ralrimiva 2626 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  U. x  e.  u
)
22 vex 2791 . . . . . . 7  |-  x  e. 
_V
2322uniex 4516 . . . . . 6  |-  U. x  e.  _V
2423elint2 3869 . . . . 5  |-  ( U. x  e.  |^| A  <->  A. u  e.  A  U. x  e.  u )
2521, 24sylibr 203 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  U. x  e.  |^| A
)
2615, 19wunpw 8329 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  ~P x  e.  u )
2726ralrimiva 2626 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  ~P x  e.  u
)
2822pwex 4193 . . . . . 6  |-  ~P x  e.  _V
2928elint2 3869 . . . . 5  |-  ( ~P x  e.  |^| A  <->  A. u  e.  A  ~P x  e.  u )
3027, 29sylibr 203 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  ~P x  e.  |^| A
)
3115adantlr 695 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  u  e. WUni )
3219adantlr 695 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  x  e.  u )
3316adantl 452 . . . . . . . . . 10  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  |^| A  C_  u )
3433sselda 3180 . . . . . . . . 9  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  /\  y  e.  |^| A )  ->  y  e.  u )
3534an32s 779 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  y  e.  u )
3631, 32, 35wunpr 8331 . . . . . . 7  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  { x ,  y }  e.  u )
3736ralrimiva 2626 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  A. u  e.  A  { x ,  y }  e.  u )
38 prex 4217 . . . . . . 7  |-  { x ,  y }  e.  _V
3938elint2 3869 . . . . . 6  |-  ( { x ,  y }  e.  |^| A  <->  A. u  e.  A  { x ,  y }  e.  u )
4037, 39sylibr 203 . . . . 5  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  { x ,  y }  e.  |^| A )
4140ralrimiva 2626 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. y  e.  |^| A { x ,  y }  e.  |^| A
)
4225, 30, 413jca 1132 . . 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 2626 . 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 447 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  =/=  (/) )
45 intex 4167 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
4644, 45sylib 188 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
47 iswun 8326 . . 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 15 . 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 1135 1  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {cpr 3641   U.cuni 3827   |^|cint 3862   Tr wtr 4113  WUnicwun 8322
This theorem is referenced by:  wunccl  8366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-int 3863  df-tr 4114  df-wun 8324
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