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Theorem gruen 8688
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 7118 . . . . 5  |-  ( B 
~~  A  <->  E. y 
y : B -1-1-onto-> A )
2 f1ofo 5682 . . . . . . . . 9  |-  ( y : B -1-1-onto-> A  ->  y : B -onto-> A )
3 simp3l 986 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  y : B -onto-> A )
4 forn 5657 . . . . . . . . . . . . 13  |-  ( y : B -onto-> A  ->  ran  y  =  A
)
53, 4syl 16 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  =  A )
6 fof 5654 . . . . . . . . . . . . . 14  |-  ( y : B -onto-> A  -> 
y : B --> A )
7 fss 5600 . . . . . . . . . . . . . 14  |-  ( ( y : B --> A  /\  A  C_  U )  -> 
y : B --> U )
86, 7sylan 459 . . . . . . . . . . . . 13  |-  ( ( y : B -onto-> A  /\  A  C_  U )  ->  y : B --> U )
9 grurn 8677 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  y : B --> U )  ->  ran  y  e.  U
)
108, 9syl3an3 1220 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  e.  U )
115, 10eqeltrrd 2512 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  A  e.  U )
12113expia 1156 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
( y : B -onto-> A  /\  A  C_  U
)  ->  A  e.  U ) )
1312exp3a 427 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -onto-> A  ->  ( A  C_  U  ->  A  e.  U ) ) )
142, 13syl5 31 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -1-1-onto-> A  -> 
( A  C_  U  ->  A  e.  U ) ) )
1514exlimdv 1647 . . . . . . 7  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  ( E. y  y : B
-1-1-onto-> A  ->  ( A  C_  U  ->  A  e.  U
) ) )
1615com3r 76 . . . . . 6  |-  ( A 
C_  U  ->  (
( U  e.  Univ  /\  B  e.  U )  ->  ( E. y 
y : B -1-1-onto-> A  ->  A  e.  U )
) )
1716expdimp 428 . . . . 5  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( B  e.  U  ->  ( E. y  y : B -1-1-onto-> A  ->  A  e.  U ) ) )
181, 17syl7bi 223 . . . 4  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( B  e.  U  ->  ( B  ~~  A  ->  A  e.  U ) ) )
1918imp3a 422 . . 3  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( ( B  e.  U  /\  B  ~~  A )  ->  A  e.  U ) )
2019ancoms 441 . 2  |-  ( ( U  e.  Univ  /\  A  C_  U )  ->  (
( B  e.  U  /\  B  ~~  A )  ->  A  e.  U
) )
21203impia 1151 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 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    C_ wss 3321   class class class wbr 4213   ran crn 4880   -->wf 5451   -onto->wfo 5453   -1-1-onto->wf1o 5454    ~~ cen 7107   Univcgru 8666
This theorem is referenced by:  grudomon  8693  gruina  8694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-tr 4304  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-en 7111  df-gru 8667
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