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Theorem gruen 9179
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 7515 . . . . 5  |-  ( B 
~~  A  <->  E. y 
y : B -1-1-onto-> A )
2 f1ofo 5814 . . . . . . . . 9  |-  ( y : B -1-1-onto-> A  ->  y : B -onto-> A )
3 simp3l 1019 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  y : B -onto-> A )
4 forn 5789 . . . . . . . . . . . . 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 5786 . . . . . . . . . . . . . 14  |-  ( y : B -onto-> A  -> 
y : B --> A )
7 fss 5730 . . . . . . . . . . . . . 14  |-  ( ( y : B --> A  /\  A  C_  U )  -> 
y : B --> U )
86, 7sylan 471 . . . . . . . . . . . . 13  |-  ( ( y : B -onto-> A  /\  A  C_  U )  ->  y : B --> U )
9 grurn 9168 . . . . . . . . . . . . 13  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  y : B --> U )  ->  ran  y  e.  U
)
108, 9syl3an3 1258 . . . . . . . . . . . 12  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  ran  y  e.  U )
115, 10eqeltrrd 2549 . . . . . . . . . . 11  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  (
y : B -onto-> A  /\  A  C_  U ) )  ->  A  e.  U )
12113expia 1193 . . . . . . . . . 10  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
( y : B -onto-> A  /\  A  C_  U
)  ->  A  e.  U ) )
1312expd 436 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -onto-> A  ->  ( A  C_  U  ->  A  e.  U ) ) )
142, 13syl5 32 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  B  e.  U )  ->  (
y : B -1-1-onto-> A  -> 
( A  C_  U  ->  A  e.  U ) ) )
1514exlimdv 1695 . . . . . . 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 437 . . . . 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 431 . . 3  |-  ( ( A  C_  U  /\  U  e.  Univ )  -> 
( ( B  e.  U  /\  B  ~~  A )  ->  A  e.  U ) )
2019ancoms 453 . 2  |-  ( ( U  e.  Univ  /\  A  C_  U )  ->  (
( B  e.  U  /\  B  ~~  A )  ->  A  e.  U
) )
21203impia 1188 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 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762    C_ wss 3469   class class class wbr 4440   ran crn 4993   -->wf 5575   -onto->wfo 5577   -1-1-onto->wf1o 5578    ~~ cen 7503   Univcgru 9157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-en 7507  df-gru 9158
This theorem is referenced by:  grudomon  9184  gruina  9185
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