Theorem nneneq 7111

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 4112	. . . . . 6 |- ( x = (/) -> ( x ~~ z <-> (/) ~~ z ) )
2		eqeq1 2239	. . . . . 6 |- ( x = (/) -> ( x = z <-> (/) = z ) )
3	1, 2	imbi12d 234	. . . . 5 |- ( x = (/) -> ( ( x ~~ z -> x = z ) <-> ( (/) ~~ z -> (/) = z ) ) )
4	3	ralbidv 2542	. . . 4 |- ( x = (/) -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( (/) ~~ z -> (/) = z ) ) )
5		breq1 4112	. . . . . 6 |- ( x = y -> ( x ~~ z <-> y ~~ z ) )
6		eqeq1 2239	. . . . . 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 2542	. . . 4 |- ( x = y -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( y ~~ z -> y = z ) ) )
9		breq1 4112	. . . . . 6 |- ( x = suc y -> ( x ~~ z <-> suc y ~~ z ) )
10		eqeq1 2239	. . . . . 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 2542	. . . 4 |- ( x = suc y -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( suc y ~~ z -> suc y = z ) ) )
13		breq1 4112	. . . . . 6 |- ( x = A -> ( x ~~ z <-> A ~~ z ) )
14		eqeq1 2239	. . . . . 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 2542	. . . 4 |- ( x = A -> ( A. z e. om ( x ~~ z -> x = z ) <-> A. z e. om ( A ~~ z -> A = z ) ) )
17		ensym 7021	. . . . . 6 |- ( (/) ~~ z -> z ~~ (/) )
18		en0 7035	. . . . . . 7 |- ( z ~~ (/) <-> z = (/) )
19		eqcom 2234	. . . . . . 7 |- ( z = (/) <-> (/) = z )
20	18, 19	bitri 184	. . . . . 6 |- ( z ~~ (/) <-> (/) = z )
21	17, 20	sylib 122	. . . . 5 |- ( (/) ~~ z -> (/) = z )
22	21	rgenw 2597	. . . 4 |- A. z e. om ( (/) ~~ z -> (/) = z )
23		nn0suc 4726	. . . . . . 7 |- ( w e. om -> ( w = (/) \/ E. z e. om w = suc z ) )
24		en0 7035	. . . . . . . . . . . 12 |- ( suc y ~~ (/) <-> suc y = (/) )
25		breq2 4113	. . . . . . . . . . . . 13 |- ( w = (/) -> ( suc y ~~ w <-> suc y ~~ (/) ) )
26		eqeq2 2242	. . . . . . . . . . . . 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 1577	. . . . . . . . . . 11 |- F/ z y e. om
32		nfra1 2573	. . . . . . . . . . 11 |- F/ z A. z e. om ( y ~~ z -> y = z )
33	31, 32	nfan 1614	. . . . . . . . . 10 |- F/ z ( y e. om /\ A. z e. om ( y ~~ z -> y = z ) )
34		nfv 1577	. . . . . . . . . 10 |- F/ z ( suc y ~~ w -> suc y = w )
35		rsp 2589	. . . . . . . . . . . . . 14 |- ( A. z e. om ( y ~~ z -> y = z ) -> ( z e. om -> ( y ~~ z -> y = z ) ) )
36		vex 2816	. . . . . . . . . . . . . . . . . 18 |- y e. _V
37		vex 2816	. . . . . . . . . . . . . . . . . 18 |- z e. _V
38	36, 37	phplem4 7109	. . . . . . . . . . . . . . . . 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 4523	. . . . . . . . . . . 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 4113	. . . . . . . . . . . . 13 |- ( w = suc z -> ( suc y ~~ w <-> suc y ~~ suc z ) )
47		eqeq2 2242	. . . . . . . . . . . . 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 2657	. . . . . . . . 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 725	. . . . . . . 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 2621	. . . . 5 |- ( y e. om -> ( A. z e. om ( y ~~ z -> y = z ) -> A. w e. om ( suc y ~~ w -> suc y = w ) ) )
56		breq2 4113	. . . . . . 7 |- ( w = z -> ( suc y ~~ w <-> suc y ~~ z ) )
57		eqeq2 2242	. . . . . . 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 2778	. . . . 5 |- ( A. w e. om ( suc y ~~ w -> suc y = w ) <-> A. z e. om ( suc y ~~ z -> suc y = z ) )
60	55, 59	imbitrdi 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 4722	. . 3 |- ( A e. om -> A. z e. om ( A ~~ z -> A = z ) )
62		breq2 4113	. . . . 5 |- ( z = B -> ( A ~~ z <-> A ~~ B ) )
63		eqeq2 2242	. . . . 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 2917	. . 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 7005	. . 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 716   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   (/)c0 3508   class class class wbr 4109   suc csuc 4486   omcom 4712   ~~ 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:  findcard2  7146  findcard2s  7147  fidcen  7156  unsnfidcex  7180  unsnfidcel  7181  exmidonfinlem  7496  hashen  11147  hashunlem  11168