Theorem gruen 9634

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.

Step	Hyp	Ref	Expression
1		bren 7964	. . . . 5 |- ( B ~~ A <-> E. y y : B -1-1-onto-> A )
2		f1ofo 6144	. . . . . . . . 9 |- ( y : B -1-1-onto-> A -> y : B -onto-> A )
3		simp3l 1089	. . . . . . . . . . . . 13 |- ( ( U e. Univ /\ B e. U /\ ( y : B -onto-> A /\ A C_ U ) ) -> y : B -onto-> A )
4		forn 6118	. . . . . . . . . . . . 13 |- ( y : B -onto-> A -> ran y = A )
5	3, 4	syl 17	. . . . . . . . . . . 12 |- ( ( U e. Univ /\ B e. U /\ ( y : B -onto-> A /\ A C_ U ) ) -> ran y = A )
6		fof 6115	. . . . . . . . . . . . . 14 |- ( y : B -onto-> A -> y : B --> A )
7		fss 6056	. . . . . . . . . . . . . 14 |- ( ( y : B --> A /\ A C_ U ) -> y : B --> U )
8	6, 7	sylan 488	. . . . . . . . . . . . 13 |- ( ( y : B -onto-> A /\ A C_ U ) -> y : B --> U )
9		grurn 9623	. . . . . . . . . . . . 13 |- ( ( U e. Univ /\ B e. U /\ y : B --> U ) -> ran y e. U )
10	8, 9	syl3an3 1361	. . . . . . . . . . . 12 |- ( ( U e. Univ /\ B e. U /\ ( y : B -onto-> A /\ A C_ U ) ) -> ran y e. U )
11	5, 10	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( U e. Univ /\ B e. U /\ ( y : B -onto-> A /\ A C_ U ) ) -> A e. U )
12	11	3expia 1267	. . . . . . . . . 10 |- ( ( U e. Univ /\ B e. U ) -> ( ( y : B -onto-> A /\ A C_ U ) -> A e. U ) )
13	12	expd 452	. . . . . . . . 9 |- ( ( U e. Univ /\ B e. U ) -> ( y : B -onto-> A -> ( A C_ U -> A e. U ) ) )
14	2, 13	syl5 34	. . . . . . . 8 |- ( ( U e. Univ /\ B e. U ) -> ( y : B -1-1-onto-> A -> ( A C_ U -> A e. U ) ) )
15	14	exlimdv 1861	. . . . . . 7 |- ( ( U e. Univ /\ B e. U ) -> ( E. y y : B -1-1-onto-> A -> ( A C_ U -> A e. U ) ) )
16	15	com3r 87	. . . . . 6 |- ( A C_ U -> ( ( U e. Univ /\ B e. U ) -> ( E. y y : B -1-1-onto-> A -> A e. U ) ) )
17	16	expdimp 453	. . . . 5 |- ( ( A C_ U /\ U e. Univ ) -> ( B e. U -> ( E. y y : B -1-1-onto-> A -> A e. U ) ) )
18	1, 17	syl7bi 245	. . . 4 |- ( ( A C_ U /\ U e. Univ ) -> ( B e. U -> ( B ~~ A -> A e. U ) ) )
19	18	impd 447	. . 3 |- ( ( A C_ U /\ U e. Univ ) -> ( ( B e. U /\ B ~~ A ) -> A e. U ) )
20	19	ancoms 469	. 2 |- ( ( U e. Univ /\ A C_ U ) -> ( ( B e. U /\ B ~~ A ) -> A e. U ) )
21	20	3impia 1261	1 |- ( ( U e. Univ /\ A C_ U /\ ( B e. U /\ B ~~ A ) ) -> A e. U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ran crn 5115   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ~~ cen 7952   Univcgru 9612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-en 7956  df-gru 9613

This theorem is referenced by:  grudomon  9639  gruina  9640