Theorem suc11reg 5710

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 5707	. . . . 5 |- -. (A e. B /\ B e. A)
2		ianor 329	. . . . 5 |- (-. (A e. B /\ B e. A) <-> (-. A e. B \/ -. B e. A))
3	1, 2	mpbi 206	. . . 4 |- (-. A e. B \/ -. B e. A)
4		eleq2 1958	. . . . . . . . . . 11 |- (suc A = suc B -> (A e. suc A <-> A e. suc B))
5		sucidg 3743	. . . . . . . . . . 11 |- (A e. _V -> A e. suc A)
6	4, 5	syl5cbi 226	. . . . . . . . . 10 |- (A e. _V -> (suc A = suc B -> A e. suc B))
7		elsucg 3732	. . . . . . . . . 10 |- (A e. _V -> (A e. suc B <-> (A e. B \/ A = B)))
8	6, 7	sylibd 219	. . . . . . . . 9 |- (A e. _V -> (suc A = suc B -> (A e. B \/ A = B)))
9	8	imp 377	. . . . . . . 8 |- ((A e. _V /\ suc A = suc B) -> (A e. B \/ A = B))
10	9	ord 249	. . . . . . 7 |- ((A e. _V /\ suc A = suc B) -> (-. A e. B -> A = B))
11	10	ex 402	. . . . . 6 |- (A e. _V -> (suc A = suc B -> (-. A e. B -> A = B)))
12	11	com23 36	. . . . 5 |- (A e. _V -> (-. A e. B -> (suc A = suc B -> A = B)))
13		eleq2 1958	. . . . . . . . . . . 12 |- (suc A = suc B -> (B e. suc A <-> B e. suc B))
14		sucidg 3743	. . . . . . . . . . . 12 |- (B e. _V -> B e. suc B)
15	13, 14	syl5cbir 228	. . . . . . . . . . 11 |- (B e. _V -> (suc A = suc B -> B e. suc A))
16		elsucg 3732	. . . . . . . . . . 11 |- (B e. _V -> (B e. suc A <-> (B e. A \/ B = A)))
17	15, 16	sylibd 219	. . . . . . . . . 10 |- (B e. _V -> (suc A = suc B -> (B e. A \/ B = A)))
18	17	imp 377	. . . . . . . . 9 |- ((B e. _V /\ suc A = suc B) -> (B e. A \/ B = A))
19	18	ord 249	. . . . . . . 8 |- ((B e. _V /\ suc A = suc B) -> (-. B e. A -> B = A))
20		eqcom 1886	. . . . . . . 8 |- (B = A <-> A = B)
21	19, 20	syl6ib 229	. . . . . . 7 |- ((B e. _V /\ suc A = suc B) -> (-. B e. A -> A = B))
22	21	ex 402	. . . . . 6 |- (B e. _V -> (suc A = suc B -> (-. B e. A -> A = B)))
23	22	com23 36	. . . . 5 |- (B e. _V -> (-. B e. A -> (suc A = suc B -> A = B)))
24	12, 23	jaao 472	. . . 4 |- ((A e. _V /\ B e. _V) -> ((-. A e. B \/ -. B e. A) -> (suc A = suc B -> A = B)))
25	3, 24	mpi 55	. . 3 |- ((A e. _V /\ B e. _V) -> (suc A = suc B -> A = B))
26		nelneq 1985	. . . . 5 |- ((suc A e. _V /\ -. suc B e. _V) -> -. suc A = suc B)
27		sucexb 3890	. . . . 5 |- (A e. _V <-> suc A e. _V)
28		sucexb 3890	. . . . . 6 |- (B e. _V <-> suc B e. _V)
29	28	notbii 204	. . . . 5 |- (-. B e. _V <-> -. suc B e. _V)
30	26, 27, 29	syl2anb 504	. . . 4 |- ((A e. _V /\ -. B e. _V) -> -. suc A = suc B)
31	30	pm2.21d 94	. . 3 |- ((A e. _V /\ -. B e. _V) -> (suc A = suc B -> A = B))
32		nelneq 1985	. . . . . . 7 |- ((suc B e. _V /\ -. suc A e. _V) -> -. suc B = suc A)
33	27	notbii 204	. . . . . . 7 |- (-. A e. _V <-> -. suc A e. _V)
34	32, 28, 33	syl2anb 504	. . . . . 6 |- ((B e. _V /\ -. A e. _V) -> -. suc B = suc A)
35	34	ancoms 484	. . . . 5 |- ((-. A e. _V /\ B e. _V) -> -. suc B = suc A)
36	35	pm2.21d 94	. . . 4 |- ((-. A e. _V /\ B e. _V) -> (suc B = suc A -> A = B))
37		eqcom 1886	. . . 4 |- (suc A = suc B <-> suc B = suc A)
38	36, 37	syl5ib 223	. . 3 |- ((-. A e. _V /\ B e. _V) -> (suc A = suc B -> A = B))
39		sucprc 3740	. . . . 5 |- (-. A e. _V -> suc A = A)
40		sucprc 3740	. . . . 5 |- (-. B e. _V -> suc B = B)
41	39, 40	eqeqan12d 1901	. . . 4 |- ((-. A e. _V /\ -. B e. _V) -> (suc A = suc B <-> A = B))
42	41	biimpd 170	. . 3 |- ((-. A e. _V /\ -. B e. _V) -> (suc A = suc B -> A = B))
43	25, 31, 38, 42	4cases 832	. 2 |- (suc A = suc B -> A = B)
44		suceq 3729	. 2 |- (A = B -> suc A = suc B)
45	43, 44	impbii 174	1 |- (suc A = suc B <-> A = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  suc csuc 3659

This theorem is referenced by:  rankxpsuc 5826  bnj168 12496  bnj551 12537

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-eprel 3583  df-fr 3625  df-suc 3663

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