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Theorem gruen 8450
Description: A Grothendieck's universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruen  |-  ( ( U  e.  Univ  /\  A  C_  U  /\  ( B  e.  U  /\  B  ~~  A ) )  ->  A  e.  U )

Proof of Theorem gruen
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bren 6887 . . . . 5  |-  ( B 
~~  A  <->  E. y 
y : B -1-1-onto-> A )
2 f1ofo 5495 . . . . . . . . 9  |-  ( y : B -1-1-onto-> A  ->  y : B -onto-> A )
3 simp3l 983 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  y : B -onto-> A )
4 forn 5470 . . . . . . . . . . . . 13  |-  ( y : B -onto-> A  ->  ran  y  =  A
)
53, 4syl 15 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  =  A )
6 fof 5467 . . . . . . . . . . . . . 14  |-  ( y : B -onto-> A  -> 
y : B --> A )
7 fss 5413 . . . . . . . . . . . . . 14  |-  ( ( y : B --> A  /\  A  C_  U )  -> 
y : B --> U )
86, 7sylan 457 . . . . . . . . . . . . 13  |-  ( ( y : B -onto-> A  /\  A  C_  U )  ->  y : B --> U )
9 grurn 8439 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  y : B --> U )  ->  ran  y  e.  U
)
108, 9syl3an3 1217 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  e.  U )
115, 10eqeltrrd 2371 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  A  e.  U )
12113expia 1153 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
( y : B -onto-> A  /\  A  C_  U
)  ->  A  e.  U ) )
1312exp3a 425 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -onto-> A  ->  ( A  C_  U  ->  A  e.  U ) ) )
142, 13syl5 28 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -1-1-onto-> A  -> 
( A  C_  U  ->  A  e.  U ) ) )
1514exlimdv 1626 . . . . . . 7  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  ( E. y  y : B
-1-1-onto-> A  ->  ( A  C_  U  ->  A  e.  U
) ) )
1615com3r 73 . . . . . 6  |-  ( A 
C_  U  ->  (
( U  e.  Univ  /\  B  e.  U )  ->  ( E. y 
y : B -1-1-onto-> A  ->  A  e.  U )
) )
1716expdimp 426 . . . . 5  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( B  e.  U  ->  ( E. y  y : B -1-1-onto-> A  ->  A  e.  U ) ) )
181, 17syl7bi 221 . . . 4  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( B  e.  U  ->  ( B  ~~  A  ->  A  e.  U ) ) )
1918imp3a 420 . . 3  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( ( B  e.  U  /\  B  ~~  A )  ->  A  e.  U ) )
2019ancoms 439 . 2  |-  ( ( U  e.  Univ  /\  A  C_  U )  ->  (
( B  e.  U  /\  B  ~~  A )  ->  A  e.  U
) )
21203impia 1148 1  |-  ( ( U  e.  Univ  /\  A  C_  U  /\  ( B  e.  U  /\  B  ~~  A ) )  ->  A  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ran crn 4706   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270    ~~ cen 6876   Univcgru 8428
This theorem is referenced by:  grudomon  8455  gruina  8456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-en 6880  df-gru 8429
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