Theorem isinfinf 7154

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 6986	. . . 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 2816	. . . . 5 |- f e. _V
4		imaexg 5115	. . . . 5 |- ( f e. _V -> ( f " n ) e. _V )
5	3, 4	ax-mp 5	. . . 4 |- ( f " n ) e. _V
6		imassrn 5112	. . . . . 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 5573	. . . . . . 7 |- ( f : om -1-1-> A -> f : om --> A )
9		frn 5517	. . . . . . 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 3249	. . . . 5 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> ( f " n ) C_ A )
12		ordom 4729	. . . . . . . 8 |- Ord om
13		ordelss 4500	. . . . . . . 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 529	. . . . . 6 |- ( ( ( om ~<_ A /\ n e. om ) /\ f : om -1-1-> A ) -> n e. om )
17		f1imaeng 7032	. . . . . 6 |- ( ( f : om -1-1-> A /\ n C_ om /\ n e. om ) -> ( f " n ) ~~ n )
18	7, 15, 16, 17	syl3anc 1274	. . . . 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 3261	. . . . . 6 |- ( x = ( f " n ) -> ( x C_ A <-> ( f " n ) C_ A ) )
21		breq1 4112	. . . . . 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 2905	. . . 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 1948	. 2 |- ( ( om ~<_ A /\ n e. om ) -> E. x ( x C_ A /\ x ~~ n ) )
26	25	ralrimiva 2615	1 |- ( om ~<_ A -> A. n e. om E. x ( x C_ A /\ x ~~ n ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   class class class wbr 4109   Ord word 4483   omcom 4712   ran crn 4750   "cima 4752   -->wf 5348   -1-1->wf1 5349   ~~ cen 6973   ~<_ cdom 6974

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-er 6767  df-en 6976  df-dom 6977

This theorem is referenced by: (None)