Theorem isinffi 8818

Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8173 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
isinffi	|- ( ( -. A e. Fin /\ B e. Fin ) -> E. f f : B -1-1-> A )

Distinct variable groups:   A, f   B, f

Proof of Theorem isinffi

Dummy variables c a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ficardom 8787	. . 3 |- ( B e. Fin -> ( card ` B ) e. om )
2		isinf 8173	. . 3 |- ( -. A e. Fin -> A. a e. om E. c ( c C_ A /\ c ~~ a ) )
3		breq2 4657	. . . . . 6 |- ( a = ( card ` B ) -> ( c ~~ a <-> c ~~ ( card ` B ) ) )
4	3	anbi2d 740	. . . . 5 |- ( a = ( card ` B ) -> ( ( c C_ A /\ c ~~ a ) <-> ( c C_ A /\ c ~~ ( card ` B ) ) ) )
5	4	exbidv 1850	. . . 4 |- ( a = ( card ` B ) -> ( E. c ( c C_ A /\ c ~~ a ) <-> E. c ( c C_ A /\ c ~~ ( card ` B ) ) ) )
6	5	rspcva 3307	. . 3 |- ( ( ( card ` B ) e. om /\ A. a e. om E. c ( c C_ A /\ c ~~ a ) ) -> E. c ( c C_ A /\ c ~~ ( card ` B ) ) )
7	1, 2, 6	syl2anr 495	. 2 |- ( ( -. A e. Fin /\ B e. Fin ) -> E. c ( c C_ A /\ c ~~ ( card ` B ) ) )
8		simprr 796	. . . . . 6 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> c ~~ ( card ` B ) )
9		ficardid 8788	. . . . . . 7 |- ( B e. Fin -> ( card ` B ) ~~ B )
10	9	ad2antlr 763	. . . . . 6 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> ( card ` B ) ~~ B )
11		entr 8008	. . . . . 6 |- ( ( c ~~ ( card ` B ) /\ ( card ` B ) ~~ B ) -> c ~~ B )
12	8, 10, 11	syl2anc 693	. . . . 5 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> c ~~ B )
13	12	ensymd 8007	. . . 4 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> B ~~ c )
14		bren 7964	. . . 4 |- ( B ~~ c <-> E. f f : B -1-1-onto-> c )
15	13, 14	sylib 208	. . 3 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> E. f f : B -1-1-onto-> c )
16		f1of1 6136	. . . . . . 7 |- ( f : B -1-1-onto-> c -> f : B -1-1-> c )
17	16	adantl 482	. . . . . 6 |- ( ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) /\ f : B -1-1-onto-> c ) -> f : B -1-1-> c )
18		simplrl 800	. . . . . 6 |- ( ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) /\ f : B -1-1-onto-> c ) -> c C_ A )
19		f1ss 6106	. . . . . 6 |- ( ( f : B -1-1-> c /\ c C_ A ) -> f : B -1-1-> A )
20	17, 18, 19	syl2anc 693	. . . . 5 |- ( ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) /\ f : B -1-1-onto-> c ) -> f : B -1-1-> A )
21	20	ex 450	. . . 4 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> ( f : B -1-1-onto-> c -> f : B -1-1-> A ) )
22	21	eximdv 1846	. . 3 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> ( E. f f : B -1-1-onto-> c -> E. f f : B -1-1-> A ) )
23	15, 22	mpd 15	. 2 |- ( ( ( -. A e. Fin /\ B e. Fin ) /\ ( c C_ A /\ c ~~ ( card ` B ) ) ) -> E. f f : B -1-1-> A )
24	7, 23	exlimddv 1863	1 |- ( ( -. A e. Fin /\ B e. Fin ) -> E. f f : B -1-1-> A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   omcom 7065   ~~ cen 7952   Fincfn 7955   cardccrd 8761

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-3or 1038  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-ne 2795  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fidomtri  8819  hashdom  13168  erdsze2lem1  31185  eldioph2lem2  37324