Theorem suc11reg 4605

Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse.

Assertion

Ref	Expression
suc11reg	|- (suc A = suc B <-> A = B)

Proof of Theorem suc11reg

Step	Hyp	Ref	Expression
1		en2lp 4602	. . . . 5 |- -. (A e. B /\ B e. A)
2		ianor 305	. . . . 5 |- (-. (A e. B /\ B e. A) <-> (-. A e. B \/ -. B e. A))
3	1, 2	mpbi 189	. . . 4 |- (-. A e. B \/ -. B e. A)
4		eleq2 1535	. . . . . . . . . . 11 |- (suc A = suc B -> (A e. suc A <-> A e. suc B))
5		sucidg 3052	. . . . . . . . . . 11 |- (A e. V -> A e. suc A)
6	4, 5	syl5cbi 209	. . . . . . . . . 10 |- (A e. V -> (suc A = suc B -> A e. suc B))
7		elsucg 3036	. . . . . . . . . 10 |- (A e. V -> (A e. suc B <-> (A e. B \/ A = B)))
8	6, 7	sylibd 202	. . . . . . . . 9 |- (A e. V -> (suc A = suc B -> (A e. B \/ A = B)))
9	8	imp 350	. . . . . . . 8 |- ((A e. V /\ suc A = suc B) -> (A e. B \/ A = B))
10	9	ord 232	. . . . . . 7 |- ((A e. V /\ suc A = suc B) -> (-. A e. B -> A = B))
11	10	ex 373	. . . . . 6 |- (A e. V -> (suc A = suc B -> (-. A e. B -> A = B)))
12	11	com23 32	. . . . 5 |- (A e. V -> (-. A e. B -> (suc A = suc B -> A = B)))
13		eleq2 1535	. . . . . . . . . . . 12 |- (suc A = suc B -> (B e. suc A <-> B e. suc B))
14		sucidg 3052	. . . . . . . . . . . 12 |- (B e. V -> B e. suc B)
15	13, 14	syl5cbir 211	. . . . . . . . . . 11 |- (B e. V -> (suc A = suc B -> B e. suc A))
16		elsucg 3036	. . . . . . . . . . 11 |- (B e. V -> (B e. suc A <-> (B e. A \/ B = A)))
17	15, 16	sylibd 202	. . . . . . . . . 10 |- (B e. V -> (suc A = suc B -> (B e. A \/ B = A)))
18	17	imp 350	. . . . . . . . 9 |- ((B e. V /\ suc A = suc B) -> (B e. A \/ B = A))
19	18	ord 232	. . . . . . . 8 |- ((B e. V /\ suc A = suc B) -> (-. B e. A -> B = A))
20		eqcom 1477	. . . . . . . 8 |- (B = A <-> A = B)
21	19, 20	syl6ib 212	. . . . . . 7 |- ((B e. V /\ suc A = suc B) -> (-. B e. A -> A = B))
22	21	ex 373	. . . . . 6 |- (B e. V -> (suc A = suc B -> (-. B e. A -> A = B)))
23	22	com23 32	. . . . 5 |- (B e. V -> (-. B e. A -> (suc A = suc B -> A = B)))
24	12, 23	jaao 427	. . . 4 |- ((A e. V /\ B e. V) -> ((-. A e. B \/ -. B e. A) -> (suc A = suc B -> A = B)))
25	3, 24	mpi 44	. . 3 |- ((A e. V /\ B e. V) -> (suc A = suc B -> A = B))
26		nelneq 1561	. . . . 5 |- ((suc A e. V /\ -. suc B e. V) -> -. suc A = suc B)
27		sucexb 3048	. . . . 5 |- (A e. V <-> suc A e. V)
28		sucexb 3048	. . . . . 6 |- (B e. V <-> suc B e. V)
29	28	negbii 187	. . . . 5 |- (-. B e. V <-> -. suc B e. V)
30	26, 27, 29	syl2anb 455	. . . 4 |- ((A e. V /\ -. B e. V) -> -. suc A = suc B)
31	30	pm2.21d 78	. . 3 |- ((A e. V /\ -. B e. V) -> (suc A = suc B -> A = B))
32		nelneq 1561	. . . . . . 7 |- ((suc B e. V /\ -. suc A e. V) -> -. suc B = suc A)
33	27	negbii 187	. . . . . . 7 |- (-. A e. V <-> -. suc A e. V)
34	32, 28, 33	syl2anb 455	. . . . . 6 |- ((B e. V /\ -. A e. V) -> -. suc B = suc A)
35	34	ancoms 436	. . . . 5 |- ((-. A e. V /\ B e. V) -> -. suc B = suc A)
36	35	pm2.21d 78	. . . 4 |- ((-. A e. V /\ B e. V) -> (suc B = suc A -> A = B))
37		eqcom 1477	. . . 4 |- (suc A = suc B <-> suc B = suc A)
38	36, 37	syl5ib 206	. . 3 |- ((-. A e. V /\ B e. V) -> (suc A = suc B -> A = B))
39		sucprc 3044	. . . . 5 |- (-. A e. V -> suc A = A)
40		sucprc 3044	. . . . 5 |- (-. B e. V -> suc B = B)
41	39, 40	eqeqan12d 1490	. . . 4 |- ((-. A e. V /\ -. B e. V) -> (suc A = suc B <-> A = B))
42	41	biimpd 153	. . 3 |- ((-. A e. V /\ -. B e. V) -> (suc A = suc B -> A = B))
43	25, 31, 38, 42	4cases 758	. 2 |- (suc A = suc B -> A = B)
44		suceq 3034	. 2 |- (A = B -> suc A = suc B)
45	43, 44	impbi 157	1 |- (suc A = suc B <-> A = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  suc csuc 2950

This theorem is referenced by:  rankxpsuc 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-eprel 2832  df-fr 2917  df-suc 2954

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