Theorem phplem4on 7122

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 6983	. . . . 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 5613	. . . . . . . 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 4717	. . . . . . . . 9 |- ( B e. om -> suc B e. om )
7		nnon 4732	. . . . . . . . 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 4536	. . . . . . . 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 535	. . . . . . 7 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A e. On )
13		f1imaen2g 7033	. . . . . . 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 1275	. . . . . 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 7023	. . . . 5 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
16		eloni 4496	. . . . . . . . 9 |- ( A e. On -> Ord A )
17		orddif 4669	. . . . . . . . 9 |- ( Ord A -> A = ( suc A \ { A } ) )
18	16, 17	syl 14	. . . . . . . 8 |- ( A e. On -> A = ( suc A \ { A } ) )
19	18	imaeq2d 5101	. . . . . . 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 5615	. . . . . . . . . 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 4537	. . . . . . . . . 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 5736	. . . . . . . . 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 3337	. . . . . . 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 5111	. . . . . . . . . . 11 |- ( f " dom f ) = ran f
29	28	eqcomi 2236	. . . . . . . . . 10 |- ran f = ( f " dom f )
30		f1ofo 5621	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
31		forn 5593	. . . . . . . . . . 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 5618	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
34	33	imaeq2d 5101	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
35	29, 32, 34	3eqtr3a 2289	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
36	35	difeq1d 3336	. . . . . . . 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 5620	. . . . . . . . . 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 5436	. . . . . . . . 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 2276	. . . . . 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 2265	. . . . 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 4135	. . . 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 536	. . . . . 6 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> B e. om )
47		fnfvelrn 5809	. . . . . . . 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 2302	. . . . . . . . 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 7110	. . . . . 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 7023	. . . 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 7024	. . . 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 1948	. 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 1398   E.wex 1541   e. wcel 2203   \ cdif 3208   C_ wss 3211   {csn 3689   class class class wbr 4109   Ord word 4483   Oncon0 4484   suc csuc 4486   omcom 4712   `'ccnv 4748   dom cdm 4749   ran crn 4750   "cima 4752   Fun wfun 5346   Fn wfn 5347   -1-1->wf1 5349   -onto->wfo 5350   -1-1-onto->wf1o 5351   ` cfv 5352   ~~ cen 6973

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-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  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-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  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

This theorem is referenced by: (None)