Theorem phplem4on 6353

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 6251	. . . . 5 |- ( suc A ~~ suc B <-> E. f f : suc A -1-1-onto-> suc B )
2	1	biimpi 118	. . . 4 |- ( suc A ~~ suc B -> E. f f : suc A -1-1-onto-> suc B )
3	2	adantl 271	. . 3 |- ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) -> E. f f : suc A -1-1-onto-> suc B )
4		f1of1 5145	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -1-1-> suc B )
5	4	adantl 271	. . . . . . 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 4336	. . . . . . . . 9 |- ( B e. om -> suc B e. om )
7		nnon 4350	. . . . . . . . 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 476	. . . . . . 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 4170	. . . . . . . 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 499	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A e. On )
13		f1imaen2g 6296	. . . . . . 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 1170	. . . . . 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 6286	. . . . 5 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
16		eloni 4130	. . . . . . . . 9 |- ( A e. On -> Ord A )
17		orddif 4290	. . . . . . . . 9 |- ( Ord A -> A = ( suc A \ { A } ) )
18	16, 17	syl 14	. . . . . . . 8 |- ( A e. On -> A = ( suc A \ { A } ) )
19	18	imaeq2d 4688	. . . . . . 7 |- ( A e. On -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
20	19	ad3antrrr 475	. . . . . 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 5147	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> f Fn suc A )
22	21	adantl 271	. . . . . . . . 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 4171	. . . . . . . . . 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 5253	. . . . . . . . 9 |- ( ( f Fn suc A /\ A e. suc A ) -> { ( f ` A ) } = ( f " { A } ) )
26	22, 24, 25	syl2anc 403	. . . . . . . 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 3090	. . . . . . 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 4698	. . . . . . . . . . 11 |- ( f " dom f ) = ran f
29	28	eqcomi 2085	. . . . . . . . . 10 |- ran f = ( f " dom f )
30		f1ofo 5153	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
31		forn 5129	. . . . . . . . . . 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 5150	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
34	33	imaeq2d 4688	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
35	29, 32, 34	3eqtr3a 2137	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
36	35	difeq1d 3089	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
37	36	adantl 271	. . . . . . 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 5152	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B <-> ( f : suc A -onto-> suc B /\ Fun `' f ) )
39	38	simprbi 269	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> Fun `' f )
40		imadif 4999	. . . . . . . . 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 271	. . . . . . 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 2124	. . . . . 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 2113	. . . . 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 3809	. . . 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 500	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> B e. om )
47		fnfvelrn 5320	. . . . . . . 8 |- ( ( f Fn suc A /\ A e. suc A ) -> ( f ` A ) e. ran f )
48	22, 24, 47	syl2anc 403	. . . . . . 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 2148	. . . . . . . . 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 271	. . . . . . 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 145	. . . . . 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 6342	. . . . . 6 |- ( ( B e. om /\ ( f ` A ) e. suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
54	46, 52, 53	syl2anc 403	. . . . 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 6286	. . . 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 6287	. . . 4 |- ( ( A ~~ ( suc B \ { ( f ` A ) } ) /\ ( suc B \ { ( f ` A ) } ) ~~ B ) -> A ~~ B )
57	45, 55, 56	syl2anc 403	. . 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 1819	. 2 |- ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) -> A ~~ B )
59	58	ex 113	1 |- ( ( A e. On /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   \ cdif 2970   C_ wss 2973   {csn 3398   class class class wbr 3785   Ord word 4117   Oncon0 4118   suc csuc 4120   omcom 4331   `'ccnv 4362   dom cdm 4363   ran crn 4364   "cima 4366   Fun wfun 4916   Fn wfn 4917   -1-1->wf1 4919   -onto->wfo 4920   -1-1-onto->wf1o 4921   ` cfv 4922   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-er 6129  df-en 6245

This theorem is referenced by: (None)