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Theorem gruen 9219
Description: A Grothendieck 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 7562 . . . . 5  |-  ( B 
~~  A  <->  E. y 
y : B -1-1-onto-> A )
2 f1ofo 5805 . . . . . . . . 9  |-  ( y : B -1-1-onto-> A  ->  y : B -onto-> A )
3 simp3l 1025 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  y : B -onto-> A )
4 forn 5780 . . . . . . . . . . . . 13  |-  ( y : B -onto-> A  ->  ran  y  =  A
)
53, 4syl 17 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  =  A )
6 fof 5777 . . . . . . . . . . . . . 14  |-  ( y : B -onto-> A  -> 
y : B --> A )
7 fss 5721 . . . . . . . . . . . . . 14  |-  ( ( y : B --> A  /\  A  C_  U )  -> 
y : B --> U )
86, 7sylan 469 . . . . . . . . . . . . 13  |-  ( ( y : B -onto-> A  /\  A  C_  U )  ->  y : B --> U )
9 grurn 9208 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  y : B --> U )  ->  ran  y  e.  U
)
108, 9syl3an3 1265 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  e.  U )
115, 10eqeltrrd 2491 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  A  e.  U )
12113expia 1199 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
( y : B -onto-> A  /\  A  C_  U
)  ->  A  e.  U ) )
1312expd 434 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -onto-> A  ->  ( A  C_  U  ->  A  e.  U ) ) )
142, 13syl5 30 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -1-1-onto-> A  -> 
( A  C_  U  ->  A  e.  U ) ) )
1514exlimdv 1745 . . . . . . 7  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  ( E. y  y : B
-1-1-onto-> A  ->  ( A  C_  U  ->  A  e.  U
) ) )
1615com3r 79 . . . . . 6  |-  ( A 
C_  U  ->  (
( U  e.  Univ  /\  B  e.  U )  ->  ( E. y 
y : B -1-1-onto-> A  ->  A  e.  U )
) )
1716expdimp 435 . . . . 5  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( B  e.  U  ->  ( E. y  y : B -1-1-onto-> A  ->  A  e.  U ) ) )
181, 17syl7bi 230 . . . 4  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( B  e.  U  ->  ( B  ~~  A  ->  A  e.  U ) ) )
1918impd 429 . . 3  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( ( B  e.  U  /\  B  ~~  A )  ->  A  e.  U ) )
2019ancoms 451 . 2  |-  ( ( U  e.  Univ  /\  A  C_  U )  ->  (
( B  e.  U  /\  B  ~~  A )  ->  A  e.  U
) )
21203impia 1194 1  |-  ( ( U  e.  Univ  /\  A  C_  U  /\  ( B  e.  U  /\  B  ~~  A ) )  ->  A  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842    C_ wss 3413   class class class wbr 4394   ran crn 4823   -->wf 5564   -onto->wfo 5566   -1-1-onto->wf1o 5567    ~~ cen 7550   Univcgru 9197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-en 7554  df-gru 9198
This theorem is referenced by:  grudomon  9224  gruina  9225
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