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Theorem intwun 9016
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 457 . . . . . 6  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  C_ WUni )
21sselda 3467 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  u  e. WUni )
3 wuntr 8986 . . . . 5  |-  ( u  e. WUni  ->  Tr  u )
42, 3syl 16 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  Tr  u )
54ralrimiva 2830 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  Tr  u
)
6 trint 4511 . . 3  |-  ( A. u  e.  A  Tr  u  ->  Tr  |^| A )
75, 6syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  Tr  |^| A )
82wun0 8999 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  (/)  e.  u
)
98ralrimiva 2830 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  (/)  e.  u
)
10 0ex 4533 . . . . 5  |-  (/)  e.  _V
1110elint2 4246 . . . 4  |-  ( (/)  e.  |^| A  <->  A. u  e.  A  (/)  e.  u
)
129, 11sylibr 212 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  (/)  e.  |^| A )
13 ne0i 3754 . . 3  |-  ( (/)  e.  |^| A  ->  |^| A  =/=  (/) )
1412, 13syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
152adantlr 714 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  u  e. WUni )
16 intss1 4254 . . . . . . . . . 10  |-  ( u  e.  A  ->  |^| A  C_  u )
1716adantl 466 . . . . . . . . 9  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  |^| A  C_  u )
1817sselda 3467 . . . . . . . 8  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  /\  x  e.  |^| A
)  ->  x  e.  u )
1918an32s 802 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  x  e.  u )
2015, 19wununi 8987 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  U. x  e.  u )
2120ralrimiva 2830 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  U. x  e.  u
)
22 vex 3081 . . . . . . 7  |-  x  e. 
_V
2322uniex 6489 . . . . . 6  |-  U. x  e.  _V
2423elint2 4246 . . . . 5  |-  ( U. x  e.  |^| A  <->  A. u  e.  A  U. x  e.  u )
2521, 24sylibr 212 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  U. x  e.  |^| A
)
2615, 19wunpw 8988 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  ~P x  e.  u )
2726ralrimiva 2830 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  ~P x  e.  u
)
2822pwex 4586 . . . . . 6  |-  ~P x  e.  _V
2928elint2 4246 . . . . 5  |-  ( ~P x  e.  |^| A  <->  A. u  e.  A  ~P x  e.  u )
3027, 29sylibr 212 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  ~P x  e.  |^| A
)
3115adantlr 714 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  u  e. WUni )
3219adantlr 714 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  x  e.  u )
3316adantl 466 . . . . . . . . . 10  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  |^| A  C_  u )
3433sselda 3467 . . . . . . . . 9  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  /\  y  e.  |^| A )  ->  y  e.  u )
3534an32s 802 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  y  e.  u )
3631, 32, 35wunpr 8990 . . . . . . 7  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  { x ,  y }  e.  u )
3736ralrimiva 2830 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  A. u  e.  A  { x ,  y }  e.  u )
38 prex 4645 . . . . . . 7  |-  { x ,  y }  e.  _V
3938elint2 4246 . . . . . 6  |-  ( { x ,  y }  e.  |^| A  <->  A. u  e.  A  { x ,  y }  e.  u )
4037, 39sylibr 212 . . . . 5  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  { x ,  y }  e.  |^| A )
4140ralrimiva 2830 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. y  e.  |^| A { x ,  y }  e.  |^| A
)
4225, 30, 413jca 1168 . . 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 2830 . 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 461 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  =/=  (/) )
45 intex 4559 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
4644, 45sylib 196 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
47 iswun 8985 . . 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 1171 1  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   {cpr 3990   U.cuni 4202   |^|cint 4239   Tr wtr 4496  WUnicwun 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-pw 3973  df-sn 3989  df-pr 3991  df-uni 4203  df-int 4240  df-tr 4497  df-wun 8983
This theorem is referenced by:  wunccl  9025
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