Theorem phplem4on 7025

Description: Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)

Assertion

Ref	Expression
phplem4on	|- ( ( A e. On /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Proof of Theorem phplem4on

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6893	. . . . 5 |- ( suc A ~~ suc B <-> E. f f : suc A -1-1-onto-> suc B )
2	1	biimpi 120	. . . 4 |- ( suc A ~~ suc B -> E. f f : suc A -1-1-onto-> suc B )
3	2	adantl 277	. . 3 |- ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) -> E. f f : suc A -1-1-onto-> suc B )
4		f1of1 5570	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -1-1-> suc B )
5	4	adantl 277	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> f : suc A -1-1-> suc B )
6		peano2 4686	. . . . . . . . 9 |- ( B e. om -> suc B e. om )
7		nnon 4701	. . . . . . . . 9 |- ( suc B e. om -> suc B e. On )
8	6, 7	syl 14	. . . . . . . 8 |- ( B e. om -> suc B e. On )
9	8	ad3antlr 493	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> suc B e. On )
10		sssucid 4505	. . . . . . . 8 |- A C_ suc A
11	10	a1i 9	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A C_ suc A )
12		simplll 533	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A e. On )
13		f1imaen2g 6943	. . . . . . 7 |- ( ( ( f : suc A -1-1-> suc B /\ suc B e. On ) /\ ( A C_ suc A /\ A e. On ) ) -> ( f " A ) ~~ A )
14	5, 9, 11, 12, 13	syl22anc 1272	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) ~~ A )
15	14	ensymd 6933	. . . . 5 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
16		eloni 4465	. . . . . . . . 9 |- ( A e. On -> Ord A )
17		orddif 4638	. . . . . . . . 9 |- ( Ord A -> A = ( suc A \ { A } ) )
18	16, 17	syl 14	. . . . . . . 8 |- ( A e. On -> A = ( suc A \ { A } ) )
19	18	imaeq2d 5067	. . . . . . 7 |- ( A e. On -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
20	19	ad3antrrr 492	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
21		f1ofn 5572	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> f Fn suc A )
22	21	adantl 277	. . . . . . . . 9 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> f Fn suc A )
23		sucidg 4506	. . . . . . . . . 10 |- ( A e. On -> A e. suc A )
24	12, 23	syl 14	. . . . . . . . 9 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A e. suc A )
25		fnsnfv 5692	. . . . . . . . 9 |- ( ( f Fn suc A /\ A e. suc A ) -> { ( f ` A ) } = ( f " { A } ) )
26	22, 24, 25	syl2anc 411	. . . . . . . 8 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> { ( f ` A ) } = ( f " { A } ) )
27	26	difeq2d 3322	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( ( f " suc A ) \ { ( f ` A ) } ) = ( ( f " suc A ) \ ( f " { A } ) ) )
28		imadmrn 5077	. . . . . . . . . . 11 |- ( f " dom f ) = ran f
29	28	eqcomi 2233	. . . . . . . . . 10 |- ran f = ( f " dom f )
30		f1ofo 5578	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
31		forn 5550	. . . . . . . . . . 11 |- ( f : suc A -onto-> suc B -> ran f = suc B )
32	30, 31	syl 14	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> ran f = suc B )
33		f1odm 5575	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
34	33	imaeq2d 5067	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
35	29, 32, 34	3eqtr3a 2286	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
36	35	difeq1d 3321	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
37	36	adantl 277	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
38		dff1o3 5577	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B <-> ( f : suc A -onto-> suc B /\ Fun `' f ) )
39	38	simprbi 275	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> Fun `' f )
40		imadif 5400	. . . . . . . . 9 |- ( Fun `' f -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
41	39, 40	syl 14	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
42	41	adantl 277	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
43	27, 37, 42	3eqtr4rd 2273	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f " ( suc A \ { A } ) ) = ( suc B \ { ( f ` A ) } ) )
44	20, 43	eqtrd 2262	. . . . 5 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) = ( suc B \ { ( f ` A ) } ) )
45	15, 44	breqtrd 4108	. . . 4 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( suc B \ { ( f ` A ) } ) )
46		simpllr 534	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> B e. om )
47		fnfvelrn 5766	. . . . . . . 8 |- ( ( f Fn suc A /\ A e. suc A ) -> ( f ` A ) e. ran f )
48	22, 24, 47	syl2anc 411	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f ` A ) e. ran f )
49	31	eleq2d 2299	. . . . . . . . 9 |- ( f : suc A -onto-> suc B -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
50	30, 49	syl 14	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
51	50	adantl 277	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
52	48, 51	mpbid 147	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( f ` A ) e. suc B )
53		phplem3g 7013	. . . . . 6 |- ( ( B e. om /\ ( f ` A ) e. suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
54	46, 52, 53	syl2anc 411	. . . . 5 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
55	54	ensymd 6933	. . . 4 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> ( suc B \ { ( f ` A ) } ) ~~ B )
56		entr 6934	. . . 4 |- ( ( A ~~ ( suc B \ { ( f ` A ) } ) /\ ( suc B \ { ( f ` A ) } ) ~~ B ) -> A ~~ B )
57	45, 55, 56	syl2anc 411	. . 3 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ B )
58	3, 57	exlimddv 1945	. 2 |- ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) -> A ~~ B )
59	58	ex 115	1 |- ( ( A e. On /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   \ cdif 3194   C_ wss 3197   {csn 3666   class class class wbr 4082   Ord word 4452   Oncon0 4453   suc csuc 4455   omcom 4681   `'ccnv 4717   dom cdm 4718   ran crn 4719   "cima 4721   Fun wfun 5311   Fn wfn 5312   -1-1->wf1 5314   -onto->wfo 5315   -1-1-onto->wf1o 5316   ` cfv 5317   ~~ cen 6883

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-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  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-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  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

This theorem is referenced by: (None)