Theorem phplem4on 6925

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 6803	. . . . 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 5500	. . . . . . . 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 4628	. . . . . . . . 9 |- ( B e. om -> suc B e. om )
7		nnon 4643	. . . . . . . . 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 4447	. . . . . . . 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 6849	. . . . . . 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 1250	. . . . . 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 6839	. . . . 5 |- ( ( ( ( A e. On /\ B e. om ) /\ suc A ~~ suc B ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
16		eloni 4407	. . . . . . . . 9 |- ( A e. On -> Ord A )
17		orddif 4580	. . . . . . . . 9 |- ( Ord A -> A = ( suc A \ { A } ) )
18	16, 17	syl 14	. . . . . . . 8 |- ( A e. On -> A = ( suc A \ { A } ) )
19	18	imaeq2d 5006	. . . . . . 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 5502	. . . . . . . . . 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 4448	. . . . . . . . . 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 5617	. . . . . . . . 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 3278	. . . . . . 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 5016	. . . . . . . . . . 11 |- ( f " dom f ) = ran f
29	28	eqcomi 2197	. . . . . . . . . 10 |- ran f = ( f " dom f )
30		f1ofo 5508	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
31		forn 5480	. . . . . . . . . . 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 5505	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
34	33	imaeq2d 5006	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
35	29, 32, 34	3eqtr3a 2250	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
36	35	difeq1d 3277	. . . . . . . 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 5507	. . . . . . . . . 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 5335	. . . . . . . . 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 2237	. . . . . 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 2226	. . . . 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 4056	. . . 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 5691	. . . . . . . 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 2263	. . . . . . . . 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 6914	. . . . . 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 6839	. . . 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 6840	. . . 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 1910	. 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 1364   E.wex 1503   e. wcel 2164   \ cdif 3151   C_ wss 3154   {csn 3619   class class class wbr 4030   Ord word 4394   Oncon0 4395   suc csuc 4397   omcom 4623   `'ccnv 4659   dom cdm 4660   ran crn 4661   "cima 4663   Fun wfun 5249   Fn wfn 5250   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255   ~~ cen 6794

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6589  df-en 6797

This theorem is referenced by: (None)