Theorem nneneq 3408

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.

Assertion

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

Proof of Theorem nneneq

Step	Hyp	Ref	Expression
1		breq1 2065	. . . . . . 7 |- (x = (/) -> (x ~~ z <-> (/) ~~ z))
2		cleq1 1107	. . . . . . 7 |- (x = (/) -> (x = z <-> (/) = z))
3	1, 2	imbi12d 474	. . . . . 6 |- (x = (/) -> ((x ~~ z -> x = z) <-> ((/) ~~ z -> (/) = z)))
4	3	biraldv 1219	. . . . 5 |- (x = (/) -> (A.z e. om (x ~~ z -> x = z) <-> A.z e. om ((/) ~~ z -> (/) = z)))
5		breq1 2065	. . . . . . 7 |- (x = y -> (x ~~ z <-> y ~~ z))
6		cleq1 1107	. . . . . . 7 |- (x = y -> (x = z <-> y = z))
7	5, 6	imbi12d 474	. . . . . 6 |- (x = y -> ((x ~~ z -> x = z) <-> (y ~~ z -> y = z)))
8	7	biraldv 1219	. . . . 5 |- (x = y -> (A.z e. om (x ~~ z -> x = z) <-> A.z e. om (y ~~ z -> y = z)))
9		breq1 2065	. . . . . . 7 |- (x = suc y -> (x ~~ z <-> suc y ~~ z))
10		cleq1 1107	. . . . . . 7 |- (x = suc y -> (x = z <-> suc y = z))
11	9, 10	imbi12d 474	. . . . . 6 |- (x = suc y -> ((x ~~ z -> x = z) <-> (suc y ~~ z -> suc y = z)))
12	11	biraldv 1219	. . . . 5 |- (x = suc y -> (A.z e. om (x ~~ z -> x = z) <-> A.z e. om (suc y ~~ z -> suc y = z)))
13		breq1 2065	. . . . . . 7 |- (x = A -> (x ~~ z <-> A ~~ z))
14		cleq1 1107	. . . . . . 7 |- (x = A -> (x = z <-> A = z))
15	13, 14	imbi12d 474	. . . . . 6 |- (x = A -> ((x ~~ z -> x = z) <-> (A ~~ z -> A = z)))
16	15	biraldv 1219	. . . . 5 |- (x = A -> (A.z e. om (x ~~ z -> x = z) <-> A.z e. om (A ~~ z -> A = z)))
17		0ex 1745	. . . . . . . . 9 |- (/) e. V
18		visset 1350	. . . . . . . . 9 |- z e. V
19		ener 3313	. . . . . . . . 9 |- Er ~~
20	17, 18, 19	ersym 3209	. . . . . . . 8 |- ((/) ~~ z -> z ~~ (/))
21		en0 3328	. . . . . . . . 9 |- (z ~~ (/) <-> z = (/))
22		cleqcom 1103	. . . . . . . . 9 |- (z = (/) <-> (/) = z)
23	21, 22	bitr 151	. . . . . . . 8 |- (z ~~ (/) <-> (/) = z)
24	20, 23	sylib 173	. . . . . . 7 |- ((/) ~~ z -> (/) = z)
25	24	a1i 7	. . . . . 6 |- (z e. om -> ((/) ~~ z -> (/) = z))
26	25	rgen 1247	. . . . 5 |- A.z e. om ((/) ~~ z -> (/) = z)
27		en0 3328	. . . . . . . . . . . . 13 |- (suc y ~~ (/) <-> suc y = (/))
28		breq2 2066	. . . . . . . . . . . . . 14 |- (w = (/) -> (suc y ~~ w <-> suc y ~~ (/)))
29		cleq2 1110	. . . . . . . . . . . . . 14 |- (w = (/) -> (suc y = w <-> suc y = (/)))
30	28, 29	bibi12d 477	. . . . . . . . . . . . 13 |- (w = (/) -> ((suc y ~~ w <-> suc y = w) <-> (suc y ~~ (/) <-> suc y = (/))))
31	27, 30	mpbiri 169	. . . . . . . . . . . 12 |- (w = (/) -> (suc y ~~ w <-> suc y = w))
32	31	biimpd 135	. . . . . . . . . . 11 |- (w = (/) -> (suc y ~~ w -> suc y = w))
33	32	a1i 7	. . . . . . . . . 10 |- ((y e. om /\ A.z e. om (y ~~ z -> y = z)) -> (w = (/) -> (suc y ~~ w -> suc y = w)))
34		ax-17 925	. . . . . . . . . . . 12 |- (y e. om -> A.z y e. om)
35		hbra1 1237	. . . . . . . . . . . 12 |- (A.z e. om (y ~~ z -> y = z) -> A.zA.z e. om (y ~~ z -> y = z))
36	34, 35	hban 704	. . . . . . . . . . 11 |- ((y e. om /\ A.z e. om (y ~~ z -> y = z)) -> A.z(y e. om /\ A.z e. om (y ~~ z -> y = z)))
37		ax-17 925	. . . . . . . . . . 11 |- ((suc y ~~ w -> suc y = w) -> A.z(suc y ~~ w -> suc y = w))
38		visset 1350	. . . . . . . . . . . . . . . . . . 19 |- y e. V
39	38, 18	phplem5 3407	. . . . . . . . . . . . . . . . . 18 |- ((y e. om /\ z e. om) -> (suc y ~~ suc z -> y ~~ z))
40	39	syl4d 28	. . . . . . . . . . . . . . . . 17 |- ((y e. om /\ z e. om) -> ((y ~~ z -> y = z) -> (suc y ~~ suc z -> y = z)))
41	40	exp 291	. . . . . . . . . . . . . . . 16 |- (y e. om -> (z e. om -> ((y ~~ z -> y = z) -> (suc y ~~ suc z -> y = z))))
42	41	a2d 15	. . . . . . . . . . . . . . 15 |- (y e. om -> ((z e. om -> (y ~~ z -> y = z)) -> (z e. om -> (suc y ~~ suc z -> y = z))))
43		ra4 1243	. . . . . . . . . . . . . . 15 |- (A.z e. om (y ~~ z -> y = z) -> (z e. om -> (y ~~ z -> y = z)))
44	42, 43	syl5 22	. . . . . . . . . . . . . 14 |- (y e. om -> (A.z e. om (y ~~ z -> y = z) -> (z e. om -> (suc y ~~ suc z -> y = z))))
45	44	imp 277	. . . . . . . . . . . . 13 |- ((y e. om /\ A.z e. om (y ~~ z -> y = z)) -> (z e. om -> (suc y ~~ suc z -> y = z)))
46		suceq 2288	. . . . . . . . . . . . 13 |- (y = z -> suc y = suc z)
47	45, 46	syl8 25	. . . . . . . . . . . 12 |- ((y e. om /\ A.z e. om (y ~~ z -> y = z)) -> (z e. om -> (suc y ~~ suc z -> suc y = suc z)))
48		breq2 2066	. . . . . . . . . . . . . 14 |- (w = suc z -> (suc y ~~ w <-> suc y ~~ suc z))
49		cleq2 1110	. . . . . . . . . . . . . 14 |- (w = suc z -> (suc y = w <-> suc y = suc z))
50	48, 49	imbi12d 474	. . . . . . . . . . . . 13 |- (w = suc z -> ((suc y ~~ w -> suc y = w) <-> (suc y ~~ suc z -> suc y = suc z)))
51	50	biimprcd 138	. . . . . . . . . . . 12 |- ((suc y ~~ suc z -> suc y = suc z) -> (w = suc z -> (suc y ~~ w -> suc y = w)))
52	47, 51	syl6 23	. . . . . . . . . . 11 |- ((y e. om /\ A.z e. om (y ~~ z -> y = z)) -> (z e. om -> (w = suc z -> (suc y ~~ w -> suc y = w))))
53	36, 37, 52	r19.23ad 1285	. . . . . . . . . 10 |- ((y e. om /\ A.z e. om (y ~~ z -> y = z)) -> (E.z e. om w = suc z -> (suc y ~~ w -> suc y = w)))
54	33, 53	jaod 329	. . . . . . . . 9 |- ((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)))
55	54	exp 291	. . . . . . . 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))))
56		nn0suc 2395	. . . . . . . 8 |- (w e. om -> (w = (/) \/ E.z e. om w = suc z))
57	55, 56	syl7 24	. . . . . . 7 |- (y e. om -> (A.z e. om (y ~~ z -> y = z) -> (w e. om -> (suc y ~~ w -> suc y = w))))
58	57	r19.21adv 1262	. . . . . 6 |- (y e. om -> (A.z e. om (y ~~ z -> y = z) -> A.w e. om (suc y ~~ w -> suc y = w)))
59		breq2 2066	. . . . . . . 8 |- (w = z -> (suc y ~~ w <-> suc y ~~ z))
60		cleq2 1110	. . . . . . . 8 |- (w = z -> (suc y = w <-> suc y = z))
61	59, 60	imbi12d 474	. . . . . . 7 |- (w = z -> ((suc y ~~ w -> suc y = w) <-> (suc y ~~ z -> suc y = z)))
62	61	cbvralv 1333	. . . . . 6 |- (A.w e. om (suc y ~~ w -> suc y = w) <-> A.z e. om (suc y ~~ z -> suc y = z))
63	58, 62	syl6ib 185	. . . . 5 |- (y e. om -> (A.z e. om (y ~~ z -> y = z) -> A.z e. om (suc y ~~ z -> suc y = z)))
64	4, 8, 12, 16, 26, 63	finds 2397	. . . 4 |- (A e. om -> A.z e. om (A ~~ z -> A = z))
65		breq2 2066	. . . . . 6 |- (z = B -> (A ~~ z <-> A ~~ B))
66		cleq2 1110	. . . . . 6 |- (z = B -> (A = z <-> A = B))
67	65, 66	imbi12d 474	. . . . 5 |- (z = B -> ((A ~~ z -> A = z) <-> (A ~~ B -> A = B)))
68	67	rcla4v 1402	. . . 4 |- (A.z e. om (A ~~ z -> A = z) -> (B e. om -> (A ~~ B -> A = B)))
69	64, 68	syl 12	. . 3 |- (A e. om -> (B e. om -> (A ~~ B -> A = B)))
70	69	imp 277	. 2 |- ((A e. om /\ B e. om) -> (A ~~ B -> A = B))
71		eqeng 3296	. . 3 |- (A e. om -> (A = B -> A ~~ B))
72	71	adantr 306	. 2 |- ((A e. om /\ B e. om) -> (A = B -> A ~~ B))
73	70, 72	impbid 397	1 |- ((A e. om /\ B e. om) -> (A ~~ B <-> A = B))

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  (/)c0 1707   class class class wbr 2054  suc csuc 2201  omcom 2372   ~~ cen 3271

This theorem is referenced by:  php 3409  onomeneq 3414  nnsdomo 3417

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429 &