Theorem nneneq 6851

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 4003	. . . . . 6 |- ( x = (/) -> ( x ~~ z <-> (/) ~~ z ) )
2		eqeq1 2184	. . . . . 6 |- ( x = (/) -> ( x = z <-> (/) = z ) )
3	1, 2	imbi12d 234	. . . . 5 |- ( x = (/) -> ( ( x ~~ z -> x = z ) <-> ( (/) ~~ z -> (/) = z ) ) )
4	3	ralbidv 2477	. . . 4 |- ( x = (/) -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( (/) ~~ z -> (/) = z ) ) )
5		breq1 4003	. . . . . 6 |- ( x = y -> ( x ~~ z <-> y ~~ z ) )
6		eqeq1 2184	. . . . . 6 |- ( x = y -> ( x = z <-> y = z ) )
7	5, 6	imbi12d 234	. . . . 5 |- ( x = y -> ( ( x ~~ z -> x = z ) <-> ( y ~~ z -> y = z ) ) )
8	7	ralbidv 2477	. . . 4 |- ( x = y -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( y ~~ z -> y = z ) ) )
9		breq1 4003	. . . . . 6 |- ( x = suc y -> ( x ~~ z <-> suc y ~~ z ) )
10		eqeq1 2184	. . . . . 6 |- ( x = suc y -> ( x = z <-> suc y = z ) )
11	9, 10	imbi12d 234	. . . . 5 |- ( x = suc y -> ( ( x ~~ z -> x = z ) <-> ( suc y ~~ z -> suc y = z ) ) )
12	11	ralbidv 2477	. . . 4 |- ( x = suc y -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( suc y ~~ z -> suc y = z ) ) )
13		breq1 4003	. . . . . 6 |- ( x = A -> ( x ~~ z <-> A ~~ z ) )
14		eqeq1 2184	. . . . . 6 |- ( x = A -> ( x = z <-> A = z ) )
15	13, 14	imbi12d 234	. . . . 5 |- ( x = A -> ( ( x ~~ z -> x = z ) <-> ( A ~~ z -> A = z ) ) )
16	15	ralbidv 2477	. . . 4 |- ( x = A -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( A ~~ z -> A = z ) ) )
17		ensym 6775	. . . . . 6 |- ( (/) ~~ z -> z ~~ (/) )
18		en0 6789	. . . . . . 7 |- ( z ~~ (/) <-> z = (/) )
19		eqcom 2179	. . . . . . 7 |- ( z = (/) <-> (/) = z )
20	18, 19	bitri 184	. . . . . 6 |- ( z ~~ (/) <-> (/) = z )
21	17, 20	sylib 122	. . . . 5 |- ( (/) ~~ z -> (/) = z )
22	21	rgenw 2532	. . . 4 |- A. z e. om ( (/) ~~ z -> (/) = z )
23		nn0suc 4600	. . . . . . 7 |- ( w e. om -> ( w = (/) \/ E. z e. om w = suc z ) )
24		en0 6789	. . . . . . . . . . . 12 |- ( suc y ~~ (/) <-> suc y = (/) )
25		breq2 4004	. . . . . . . . . . . . 13 |- ( w = (/) -> ( suc y ~~ w <-> suc y ~~ (/) ) )
26		eqeq2 2187	. . . . . . . . . . . . 13 |- ( w = (/) -> ( suc y = w <-> suc y = (/) ) )
27	25, 26	bibi12d 235	. . . . . . . . . . . 12 |- ( w = (/) -> ( ( suc y ~~ w <-> suc y = w ) <-> ( suc y ~~ (/) <-> suc y = (/) ) ) )
28	24, 27	mpbiri 168	. . . . . . . . . . 11 |- ( w = (/) -> ( suc y ~~ w <-> suc y = w ) )
29	28	biimpd 144	. . . . . . . . . 10 |- ( w = (/) -> ( suc y ~~ w -> suc y = w ) )
30	29	a1i 9	. . . . . . . . 9 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( w = (/) -> ( suc y ~~ w -> suc y = w ) ) )
31		nfv 1528	. . . . . . . . . . 11 |- F/ z y e. om
32		nfra1 2508	. . . . . . . . . . 11 |- F/ z A. z e. om ( y ~~ z -> y = z )
33	31, 32	nfan 1565	. . . . . . . . . 10 |- F/ z ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) )
34		nfv 1528	. . . . . . . . . 10 |- F/ z ( suc y ~~ w -> suc y = w )
35		rsp 2524	. . . . . . . . . . . . . 14 |- ( A. z e. om ( y ~~ z -> y = z ) -> ( z e. om -> ( y ~~ z -> y = z ) ) )
36		vex 2740	. . . . . . . . . . . . . . . . . 18 |- y e. _V
37		vex 2740	. . . . . . . . . . . . . . . . . 18 |- z e. _V
38	36, 37	phplem4 6849	. . . . . . . . . . . . . . . . 17 |- ( ( y e. om /\ z e. om ) -> ( suc y ~~ suc z -> y ~~ z ) )
39	38	imim1d 75	. . . . . . . . . . . . . . . 16 |- ( ( y e. om /\ z e. om ) -> ( ( y ~~ z -> y = z ) -> ( suc y ~~ suc z -> y = z ) ) )
40	39	ex 115	. . . . . . . . . . . . . . 15 |- ( y e. om -> ( z e. om -> ( ( y ~~ z -> y = z ) -> ( suc y ~~ suc z -> y = z ) ) ) )
41	40	a2d 26	. . . . . . . . . . . . . 14 |- ( y e. om -> ( ( z e. om -> ( y ~~ z -> y = z ) ) -> ( z e. om -> ( suc y ~~ suc z -> y = z ) ) ) )
42	35, 41	syl5 32	. . . . . . . . . . . . 13 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> ( z e. om -> ( suc y ~~ suc z -> y = z ) ) ) )
43	42	imp 124	. . . . . . . . . . . 12 |- ( ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) ) -> ( z e. om -> ( suc y ~~ suc z -> y = z ) ) )
44		suceq 4399	. . . . . . . . . . . 12 |- ( y = z -> suc y = suc z )
45	43, 44	syl8 71	. . . . . . . . . . 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 4004	. . . . . . . . . . . . 13 |- ( w = suc z -> ( suc y ~~ w <-> suc y ~~ suc z ) )
47		eqeq2 2187	. . . . . . . . . . . . 13 |- ( w = suc z -> ( suc y = w <-> suc y = suc z ) )
48	46, 47	imbi12d 234	. . . . . . . . . . . 12 |- ( w = suc z -> ( ( suc y ~~ w -> suc y = w ) <-> ( suc y ~~ suc z -> suc y = suc z ) ) )
49	48	biimprcd 160	. . . . . . . . . . 11 |- ( ( suc y ~~ suc z -> suc y = suc z ) -> ( w = suc z -> ( suc y ~~ w -> suc y = w ) ) )
50	45, 49	syl6 33	. . . . . . . . . 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 2591	. . . . . . . . 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 717	. . . . . . . 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 115	. . . . . . 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 69	. . . . . 6 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> ( w e. om -> ( suc y ~~ w -> suc y = w ) ) ) )
55	54	ralrimdv 2556	. . . . 5 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> A. w e. om ( suc y ~~ w -> suc y = w ) ) )
56		breq2 4004	. . . . . . 7 |- ( w = z -> ( suc y ~~ w <-> suc y ~~ z ) )
57		eqeq2 2187	. . . . . . 7 |- ( w = z -> ( suc y = w <-> suc y = z ) )
58	56, 57	imbi12d 234	. . . . . 6 |- ( w = z -> ( ( suc y ~~ w -> suc y = w ) <-> ( suc y ~~ z -> suc y = z ) ) )
59	58	cbvralv 2703	. . . . 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 161	. . . 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 4596	. . 3 |- ( A e. om -> A. z e. om ( A ~~ z -> A = z ) )
62		breq2 4004	. . . . 5 |- ( z = B -> ( A ~~ z <-> A ~~ B ) )
63		eqeq2 2187	. . . . 5 |- ( z = B -> ( A = z <-> A = B ) )
64	62, 63	imbi12d 234	. . . 4 |- ( z = B -> ( ( A ~~ z -> A = z ) <-> ( A ~~ B -> A = B ) ) )
65	64	rspcv 2837	. . 3 |- ( B e. om -> ( A. z e. om ( A ~~ z -> A = z ) -> ( A ~~ B -> A = B ) ) )
66	61, 65	mpan9 281	. 2 |- ( ( A e. om /\ B e. om ) -> ( A ~~ B -> A = B ) )
67		eqeng 6760	. . 3 |- ( A e. om -> ( A = B -> A ~~ B ) )
68	67	adantr 276	. 2 |- ( ( A e. om /\ B e. om ) -> ( A = B -> A ~~ B ) )
69	66, 68	impbid 129	1 |- ( ( A e. om /\ B e. om ) -> ( A ~~ B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3422   class class class wbr 4000   suc csuc 4362   omcom 4586   ~~ cen 6732

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-er 6529  df-en 6735

This theorem is referenced by:  findcard2  6883  findcard2s  6884  unsnfidcex  6913  unsnfidcel  6914  exmidonfinlem  7186  hashen  10748  hashunlem  10768