MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intwun Structured version   Unicode version

Theorem intwun 9046
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 455 . . . . . 6  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  C_ WUni )
21sselda 3434 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  u  e. WUni )
3 wuntr 9016 . . . . 5  |-  ( u  e. WUni  ->  Tr  u )
42, 3syl 16 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  Tr  u )
54ralrimiva 2810 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  Tr  u
)
6 trint 4492 . . 3  |-  ( A. u  e.  A  Tr  u  ->  Tr  |^| A )
75, 6syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  Tr  |^| A )
82wun0 9029 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  (/)  e.  u
)
98ralrimiva 2810 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A. u  e.  A  (/)  e.  u
)
10 0ex 4514 . . . . 5  |-  (/)  e.  _V
1110elint2 4223 . . . 4  |-  ( (/)  e.  |^| A  <->  A. u  e.  A  (/)  e.  u
)
129, 11sylibr 212 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  (/)  e.  |^| A )
13 ne0i 3734 . . 3  |-  ( (/)  e.  |^| A  ->  |^| A  =/=  (/) )
1412, 13syl 16 . 2  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
152adantlr 712 . . . . . . 7  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  u  e. WUni )
16 intss1 4231 . . . . . . . . . 10  |-  ( u  e.  A  ->  |^| A  C_  u )
1716adantl 464 . . . . . . . . 9  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  u  e.  A )  ->  |^| A  C_  u )
1817sselda 3434 . . . . . . . 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 9017 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  U. x  e.  u )
2120ralrimiva 2810 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  U. x  e.  u
)
22 vex 3054 . . . . . . 7  |-  x  e. 
_V
2322uniex 6517 . . . . . 6  |-  U. x  e.  _V
2423elint2 4223 . . . . 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 9018 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  ~P x  e.  u )
2726ralrimiva 2810 . . . . 5  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. u  e.  A  ~P x  e.  u
)
2822pwex 4565 . . . . . 6  |-  ~P x  e.  _V
2928elint2 4223 . . . . 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 712 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  u  e. WUni )
3219adantlr 712 . . . . . . . 8  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  x  e.  u )
3316adantl 464 . . . . . . . . . 10  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  u  e.  A
)  ->  |^| A  C_  u )
3433sselda 3434 . . . . . . . . 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 9020 . . . . . . 7  |-  ( ( ( ( ( A 
C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  /\  u  e.  A )  ->  { x ,  y }  e.  u )
3736ralrimiva 2810 . . . . . 6  |-  ( ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  /\  y  e.  |^| A )  ->  A. u  e.  A  { x ,  y }  e.  u )
38 prex 4621 . . . . . . 7  |-  { x ,  y }  e.  _V
3938elint2 4223 . . . . . 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 2810 . . . 4  |-  ( ( ( A  C_ WUni  /\  A  =/=  (/) )  /\  x  e.  |^| A )  ->  A. y  e.  |^| A { x ,  y }  e.  |^| A
)
4225, 30, 413jca 1174 . . 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 2810 . 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 459 . . . 4  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  A  =/=  (/) )
45 intex 4538 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
4644, 45sylib 196 . . 3  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e.  _V )
47 iswun 9015 . . 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 1177 1  |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    e. wcel 1836    =/= wne 2591   A.wral 2746   _Vcvv 3051    C_ wss 3406   (/)c0 3728   ~Pcpw 3944   {cpr 3963   U.cuni 4180   |^|cint 4216   Tr wtr 4477  WUnicwun 9011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-pw 3946  df-sn 3962  df-pr 3964  df-uni 4181  df-int 4217  df-tr 4478  df-wun 9013
This theorem is referenced by:  wunccl  9055
  Copyright terms: Public domain W3C validator