Theorem infpssr 9130

Description: Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
infpssr	|- ( ( X C. A /\ X ~~ A ) -> om ~<_ A )

Proof of Theorem infpssr

Dummy variables y f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pssnel 4039	. . 3 |- ( X C. A -> E. y ( y e. A /\ -. y e. X ) )
2	1	adantr 481	. 2 |- ( ( X C. A /\ X ~~ A ) -> E. y ( y e. A /\ -. y e. X ) )
3		eldif 3584	. . . 4 |- ( y e. ( A \ X ) <-> ( y e. A /\ -. y e. X ) )
4		pssss 3702	. . . . . 6 |- ( X C. A -> X C_ A )
5		bren 7964	. . . . . . . 8 |- ( X ~~ A <-> E. f f : X -1-1-onto-> A )
6		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> f : X -1-1-onto-> A )
7		f1ofo 6144	. . . . . . . . . . . . 13 |- ( f : X -1-1-onto-> A -> f : X -onto-> A )
8		forn 6118	. . . . . . . . . . . . 13 |- ( f : X -onto-> A -> ran f = A )
9	6, 7, 8	3syl 18	. . . . . . . . . . . 12 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> ran f = A )
10		vex 3203	. . . . . . . . . . . . 13 |- f e. _V
11	10	rnex 7100	. . . . . . . . . . . 12 |- ran f e. _V
12	9, 11	syl6eqelr 2710	. . . . . . . . . . 11 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> A e. _V )
13		simplr 792	. . . . . . . . . . . 12 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> X C_ A )
14		simpll 790	. . . . . . . . . . . 12 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> y e. ( A \ X ) )
15		eqid 2622	. . . . . . . . . . . 12 |- ( rec ( `' f , y ) |` om ) = ( rec ( `' f , y ) |` om )
16	13, 6, 14, 15	infpssrlem5 9129	. . . . . . . . . . 11 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> ( A e. _V -> om ~<_ A ) )
17	12, 16	mpd 15	. . . . . . . . . 10 |- ( ( ( y e. ( A \ X ) /\ X C_ A ) /\ f : X -1-1-onto-> A ) -> om ~<_ A )
18	17	ex 450	. . . . . . . . 9 |- ( ( y e. ( A \ X ) /\ X C_ A ) -> ( f : X -1-1-onto-> A -> om ~<_ A ) )
19	18	exlimdv 1861	. . . . . . . 8 |- ( ( y e. ( A \ X ) /\ X C_ A ) -> ( E. f f : X -1-1-onto-> A -> om ~<_ A ) )
20	5, 19	syl5bi 232	. . . . . . 7 |- ( ( y e. ( A \ X ) /\ X C_ A ) -> ( X ~~ A -> om ~<_ A ) )
21	20	ex 450	. . . . . 6 |- ( y e. ( A \ X ) -> ( X C_ A -> ( X ~~ A -> om ~<_ A ) ) )
22	4, 21	syl5 34	. . . . 5 |- ( y e. ( A \ X ) -> ( X C. A -> ( X ~~ A -> om ~<_ A ) ) )
23	22	impd 447	. . . 4 |- ( y e. ( A \ X ) -> ( ( X C. A /\ X ~~ A ) -> om ~<_ A ) )
24	3, 23	sylbir 225	. . 3 |- ( ( y e. A /\ -. y e. X ) -> ( ( X C. A /\ X ~~ A ) -> om ~<_ A ) )
25	24	exlimiv 1858	. 2 |- ( E. y ( y e. A /\ -. y e. X ) -> ( ( X C. A /\ X ~~ A ) -> om ~<_ A ) )
26	2, 25	mpcom 38	1 |- ( ( X C. A /\ X ~~ A ) -> om ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   C. wpss 3575   class class class wbr 4653   `'ccnv 5113   ran crn 5115   |` cres 5116   -onto->wfo 5886   -1-1-onto->wf1o 5887   omcom 7065   reccrdg 7505   ~~ cen 7952   ~<_ cdom 7953

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-pred 5680  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-en 7956  df-dom 7957

This theorem is referenced by:  isfin4-2  9136