Theorem nneneq 8143

Description: Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)

Assertion

Ref	Expression
nneneq	|- ( ( A e. om /\ B e. om ) -> ( A ~~ B <-> A = B ) )

Proof of Theorem nneneq

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . . 6 |- ( x = (/) -> ( x ~~ z <-> (/) ~~ z ) )
2		eqeq1 2626	. . . . . 6 |- ( x = (/) -> ( x = z <-> (/) = z ) )
3	1, 2	imbi12d 334	. . . . 5 |- ( x = (/) -> ( ( x ~~ z -> x = z ) <-> ( (/) ~~ z -> (/) = z ) ) )
4	3	ralbidv 2986	. . . 4 |- ( x = (/) -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( (/) ~~ z -> (/) = z ) ) )
5		breq1 4656	. . . . . 6 |- ( x = y -> ( x ~~ z <-> y ~~ z ) )
6		eqeq1 2626	. . . . . 6 |- ( x = y -> ( x = z <-> y = z ) )
7	5, 6	imbi12d 334	. . . . 5 |- ( x = y -> ( ( x ~~ z -> x = z ) <-> ( y ~~ z -> y = z ) ) )
8	7	ralbidv 2986	. . . 4 |- ( x = y -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( y ~~ z -> y = z ) ) )
9		breq1 4656	. . . . . 6 |- ( x = suc y -> ( x ~~ z <-> suc y ~~ z ) )
10		eqeq1 2626	. . . . . 6 |- ( x = suc y -> ( x = z <-> suc y = z ) )
11	9, 10	imbi12d 334	. . . . 5 |- ( x = suc y -> ( ( x ~~ z -> x = z ) <-> ( suc y ~~ z -> suc y = z ) ) )
12	11	ralbidv 2986	. . . 4 |- ( x = suc y -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( suc y ~~ z -> suc y = z ) ) )
13		breq1 4656	. . . . . 6 |- ( x = A -> ( x ~~ z <-> A ~~ z ) )
14		eqeq1 2626	. . . . . 6 |- ( x = A -> ( x = z <-> A = z ) )
15	13, 14	imbi12d 334	. . . . 5 |- ( x = A -> ( ( x ~~ z -> x = z ) <-> ( A ~~ z -> A = z ) ) )
16	15	ralbidv 2986	. . . 4 |- ( x = A -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( A ~~ z -> A = z ) ) )
17		ensym 8005	. . . . . 6 |- ( (/) ~~ z -> z ~~ (/) )
18		en0 8019	. . . . . . 7 |- ( z ~~ (/) <-> z = (/) )
19		eqcom 2629	. . . . . . 7 |- ( z = (/) <-> (/) = z )
20	18, 19	bitri 264	. . . . . 6 |- ( z ~~ (/) <-> (/) = z )
21	17, 20	sylib 208	. . . . 5 |- ( (/) ~~ z -> (/) = z )
22	21	rgenw 2924	. . . 4 |- A. z e. om ( (/) ~~ z -> (/) = z )
23		nn0suc 7090	. . . . . . 7 |- ( w e. om -> ( w = (/) \/ E. z e. om w = suc z ) )
24		en0 8019	. . . . . . . . . . . 12 |- ( suc y ~~ (/) <-> suc y = (/) )
25		breq2 4657	. . . . . . . . . . . . 13 |- ( w = (/) -> ( suc y ~~ w <-> suc y ~~ (/) ) )
26		eqeq2 2633	. . . . . . . . . . . . 13 |- ( w = (/) -> ( suc y = w <-> suc y = (/) ) )
27	25, 26	bibi12d 335	. . . . . . . . . . . 12 |- ( w = (/) -> ( ( suc y ~~ w <-> suc y = w ) <-> ( suc y ~~ (/) <-> suc y = (/) ) ) )
28	24, 27	mpbiri 248	. . . . . . . . . . 11 |- ( w = (/) -> ( suc y ~~ w <-> suc y = w ) )
29	28	biimpd 219	. . . . . . . . . 10 |- ( w = (/) -> ( suc y ~~ w -> suc y = w ) )
30	29	a1i 11	. . . . . . . . 9 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( w = (/) -> ( suc y ~~ w -> suc y = w ) ) )
31		nfv 1843	. . . . . . . . . . 11 |- F/ z y e. om
32		nfra1 2941	. . . . . . . . . . 11 |- F/ z A. z e. om ( y ~~ z -> y = z )
33	31, 32	nfan 1828	. . . . . . . . . 10 |- F/ z ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) )
34		nfv 1843	. . . . . . . . . 10 |- F/ z ( suc y ~~ w -> suc y = w )
35		rsp 2929	. . . . . . . . . . . . . 14 |- ( A. z e. om ( y ~~ z -> y = z ) -> ( z e. om -> ( y ~~ z -> y = z ) ) )
36		vex 3203	. . . . . . . . . . . . . . . . . 18 |- y e. _V
37		vex 3203	. . . . . . . . . . . . . . . . . 18 |- z e. _V
38	36, 37	phplem4 8142	. . . . . . . . . . . . . . . . 17 |- ( ( y e. om /\ z e. om ) -> ( suc y ~~ suc z -> y ~~ z ) )
39	38	imim1d 82	. . . . . . . . . . . . . . . 16 |- ( ( y e. om /\ z e. om ) -> ( ( y ~~ z -> y = z ) -> ( suc y ~~ suc z -> y = z ) ) )
40	39	ex 450	. . . . . . . . . . . . . . 15 |- ( y e. om -> ( z e. om -> ( ( y ~~ z -> y = z ) -> ( suc y ~~ suc z -> y = z ) ) ) )
41	40	a2d 29	. . . . . . . . . . . . . 14 |- ( y e. om -> ( ( z e. om -> ( y ~~ z -> y = z ) ) -> ( z e. om -> ( suc y ~~ suc z -> y = z ) ) ) )
42	35, 41	syl5 34	. . . . . . . . . . . . 13 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> ( z e. om -> ( suc y ~~ suc z -> y = z ) ) ) )
43	42	imp 445	. . . . . . . . . . . 12 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( z e. om -> ( suc y ~~ suc z -> y = z ) ) )
44		suceq 5790	. . . . . . . . . . . 12 |- ( y = z -> suc y = suc z )
45	43, 44	syl8 76	. . . . . . . . . . 11 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( z e. om -> ( suc y ~~ suc z -> suc y = suc z ) ) )
46		breq2 4657	. . . . . . . . . . . . 13 |- ( w = suc z -> ( suc y ~~ w <-> suc y ~~ suc z ) )
47		eqeq2 2633	. . . . . . . . . . . . 13 |- ( w = suc z -> ( suc y = w <-> suc y = suc z ) )
48	46, 47	imbi12d 334	. . . . . . . . . . . 12 |- ( w = suc z -> ( ( suc y ~~ w -> suc y = w ) <-> ( suc y ~~ suc z -> suc y = suc z ) ) )
49	48	biimprcd 240	. . . . . . . . . . 11 |- ( ( suc y ~~ suc z -> suc y = suc z ) -> ( w = suc z -> ( suc y ~~ w -> suc y = w ) ) )
50	45, 49	syl6 35	. . . . . . . . . 10 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( z e. om -> ( w = suc z -> ( suc y ~~ w -> suc y = w ) ) ) )
51	33, 34, 50	rexlimd 3026	. . . . . . . . 9 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( E. z e. om w = suc z -> ( suc y ~~ w -> suc y = w ) ) )
52	30, 51	jaod 395	. . . . . . . 8 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( ( w = (/) \/ E. z e. om w = suc z ) -> ( suc y ~~ w -> suc y = w ) ) )
53	52	ex 450	. . . . . . 7 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> ( ( w = (/) \/ E. z e. om w = suc z ) -> ( suc y ~~ w -> suc y = w ) ) ) )
54	23, 53	syl7 74	. . . . . 6 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> ( w e. om -> ( suc y ~~ w -> suc y = w ) ) ) )
55	54	ralrimdv 2968	. . . . 5 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> A. w e. om ( suc y ~~ w -> suc y = w ) ) )
56		breq2 4657	. . . . . . 7 |- ( w = z -> ( suc y ~~ w <-> suc y ~~ z ) )
57		eqeq2 2633	. . . . . . 7 |- ( w = z -> ( suc y = w <-> suc y = z ) )
58	56, 57	imbi12d 334	. . . . . 6 |- ( w = z -> ( ( suc y ~~ w -> suc y = w ) <-> ( suc y ~~ z -> suc y = z ) ) )
59	58	cbvralv 3171	. . . . 5 |- ( A. w e. om ( suc y ~~ w -> suc y = w ) <-> A. z e. om ( suc y ~~ z -> suc y = z ) )
60	55, 59	syl6ib 241	. . . 4 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> A. z e. om ( suc y ~~ z -> suc y = z ) ) )
61	4, 8, 12, 16, 22, 60	finds 7092	. . 3 |- ( A e. om -> A. z e. om ( A ~~ z -> A = z ) )
62		breq2 4657	. . . . 5 |- ( z = B -> ( A ~~ z <-> A ~~ B ) )
63		eqeq2 2633	. . . . 5 |- ( z = B -> ( A = z <-> A = B ) )
64	62, 63	imbi12d 334	. . . 4 |- ( z = B -> ( ( A ~~ z -> A = z ) <-> ( A ~~ B -> A = B ) ) )
65	64	rspcv 3305	. . 3 |- ( B e. om -> ( A. z e. om ( A ~~ z -> A = z ) -> ( A ~~ B -> A = B ) ) )
66	61, 65	mpan9 486	. 2 |- ( ( A e. om /\ B e. om ) -> ( A ~~ B -> A = B ) )
67		eqeng 7989	. . 3 |- ( A e. om -> ( A = B -> A ~~ B ) )
68	67	adantr 481	. 2 |- ( ( A e. om /\ B e. om ) -> ( A = B -> A ~~ B ) )
69	66, 68	impbid 202	1 |- ( ( A e. om /\ B e. om ) -> ( A ~~ B <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   (/)c0 3915   class class class wbr 4653   suc csuc 5725   omcom 7065   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-er 7742  df-en 7956

This theorem is referenced by:  php  8144  onomeneq  8150  nnsdomo  8155  fineqvlem  8174  dif1en  8193  findcard2  8200  cardnn  8789