Theorem isinfinf 7055

Description: An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)

Assertion

Ref	Expression
isinfinf	|- ( om ~<_ A -> A. n e. om E. x ( x C_ A /\ x ~~ n ) )

Distinct variable group:   A, n, x

Proof of Theorem isinfinf

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6896	. . . 4 |- ( om ~<_ A -> E. f f : om -1-1-> A )
2	1	adantr 276	. . 3 |- ( ( om ~<_ A /\ n e. om ) -> E. f f : om -1-1-> A )
3		vex 2802	. . . . 5 |- f e. _V
4		imaexg 5081	. . . . 5 |- ( f e. _V -> ( f " n ) e. _V )
5	3, 4	ax-mp 5	. . . 4 |- ( f " n ) e. _V
6		imassrn 5078	. . . . . 6 |- ( f " n ) C_ ran f
7		simpr 110	. . . . . . 7 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> f : om -1-1-> A )
8		f1f 5530	. . . . . . 7 |- ( f : om -1-1-> A -> f : om --> A )
9		frn 5481	. . . . . . 7 |- ( f : om --> A -> ran f C_ A )
10	7, 8, 9	3syl 17	. . . . . 6 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> ran f C_ A )
11	6, 10	sstrid 3235	. . . . 5 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> ( f " n ) C_ A )
12		ordom 4698	. . . . . . . 8 |- Ord om
13		ordelss 4469	. . . . . . . 8 |- ( ( Ord om /\ n e. om ) -> n C_ om )
14	12, 13	mpan 424	. . . . . . 7 |- ( n e. om -> n C_ om )
15	14	ad2antlr 489	. . . . . 6 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> n C_ om )
16		simplr 528	. . . . . 6 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> n e. om )
17		f1imaeng 6942	. . . . . 6 |- ( ( f : om -1-1-> A /\ n C_ om /\ n e. om ) -> ( f " n ) ~~ n )
18	7, 15, 16, 17	syl3anc 1271	. . . . 5 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> ( f " n ) ~~ n )
19	11, 18	jca 306	. . . 4 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> ( ( f " n ) C_ A /\ ( f " n ) ~~ n ) )
20		sseq1 3247	. . . . . 6 |- ( x = ( f " n ) -> ( x C_ A <-> ( f " n ) C_ A ) )
21		breq1 4085	. . . . . 6 |- ( x = ( f " n ) -> ( x ~~ n <-> ( f " n ) ~~ n ) )
22	20, 21	anbi12d 473	. . . . 5 |- ( x = ( f " n ) -> ( ( x C_ A /\ x ~~ n ) <-> ( ( f " n ) C_ A /\ ( f " n ) ~~ n ) ) )
23	22	spcegv 2891	. . . 4 |- ( ( f " n ) e. _V -> ( ( ( f " n ) C_ A /\ ( f " n ) ~~ n ) -> E. x ( x C_ A /\ x ~~ n ) ) )
24	5, 19, 23	mpsyl 65	. . 3 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> E. x ( x C_ A /\ x ~~ n ) )
25	2, 24	exlimddv 1945	. 2 |- ( ( om ~<_ A /\ n e. om ) -> E. x ( x C_ A /\ x ~~ n ) )
26	25	ralrimiva 2603	1 |- ( om ~<_ A -> A. n e. om E. x ( x C_ A /\ x ~~ n ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   class class class wbr 4082   Ord word 4452   omcom 4681   ran crn 4719   "cima 4721   -->wf 5313   -1-1->wf1 5314   ~~ cen 6883   ~<_ cdom 6884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886  df-dom 6887

This theorem is referenced by: (None)