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Theorem intwun 9178
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 464 . . . . . 6  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  C_ WUni )
21sselda 3418 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  u  e. WUni )
3 wuntr 9148 . . . . 5  |-  ( u  e. WUni  ->  Tr  u )
42, 3syl 17 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  Tr  u )
54ralrimiva 2809 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  Tr  u
)
6 trint 4505 . . 3  |-  ( A. u  e.  A  Tr  u  ->  Tr  |^| A )
75, 6syl 17 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  Tr  |^| A )
82wun0 9161 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  (/)  e.  u
)
98ralrimiva 2809 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  (/)  e.  u
)
10 0ex 4528 . . . . 5  |-  (/)  e.  _V
1110elint2 4233 . . . 4  |-  ( (/)  e.  |^| A  <->  A. u  e.  A  (/)  e.  u
)
129, 11sylibr 217 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  (/)  e.  |^| A )
13 ne0i 3728 . . 3  |-  ( (/)  e.  |^| A  ->  |^| A  =/=  (/) )
1412, 13syl 17 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
152adantlr 729 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  u  e. WUni )
16 intss1 4241 . . . . . . . . . 10  |-  ( u  e.  A  ->  |^| A  C_  u )
1716adantl 473 . . . . . . . . 9  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  |^| A  C_  u )
1817sselda 3418 . . . . . . . 8  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  /\  x  e.  |^| A
)  ->  x  e.  u )
1918an32s 821 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  x  e.  u )
2015, 19wununi 9149 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  U. x  e.  u )
2120ralrimiva 2809 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  U. x  e.  u
)
22 vex 3034 . . . . . . 7  |-  x  e. 
_V
2322uniex 6606 . . . . . 6  |-  U. x  e.  _V
2423elint2 4233 . . . . 5  |-  ( U. x  e.  |^| A  <->  A. u  e.  A  U. x  e.  u )
2521, 24sylibr 217 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  U. x  e.  |^| A
)
2615, 19wunpw 9150 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  ~P x  e.  u )
2726ralrimiva 2809 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  ~P x  e.  u
)
2822pwex 4584 . . . . . 6  |-  ~P x  e.  _V
2928elint2 4233 . . . . 5  |-  ( ~P x  e.  |^| A  <->  A. u  e.  A  ~P x  e.  u )
3027, 29sylibr 217 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  ~P x  e.  |^| A
)
3115adantlr 729 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  u  e. WUni )
3219adantlr 729 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  x  e.  u )
3316adantl 473 . . . . . . . . . 10  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  |^| A  C_  u )
3433sselda 3418 . . . . . . . . 9  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  /\  y  e.  |^| A )  ->  y  e.  u )
3534an32s 821 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  y  e.  u )
3631, 32, 35wunpr 9152 . . . . . . 7  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  { x ,  y }  e.  u )
3736ralrimiva 2809 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  A. u  e.  A  { x ,  y }  e.  u )
38 prex 4642 . . . . . . 7  |-  { x ,  y }  e.  _V
3938elint2 4233 . . . . . 6  |-  ( { x ,  y }  e.  |^| A  <->  A. u  e.  A  { x ,  y }  e.  u )
4037, 39sylibr 217 . . . . 5  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  { x ,  y }  e.  |^| A )
4140ralrimiva 2809 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. y  e.  |^| A { x ,  y }  e.  |^| A
)
4225, 30, 413jca 1210 . . 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 2809 . 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 468 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  =/=  (/) )
45 intex 4557 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
4644, 45sylib 201 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
47 iswun 9147 . . 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 17 . 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 1213 1  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {cpr 3961   U.cuni 4190   |^|cint 4226   Tr wtr 4490  WUnicwun 9143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-pw 3944  df-sn 3960  df-pr 3962  df-uni 4191  df-int 4227  df-tr 4491  df-wun 9145
This theorem is referenced by:  wunccl  9187
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