Theorem suc11reg 4614

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 4611	. . . . 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 1538	. . . . . . . . . . 11 |- (suc A = suc B -> (A e. suc A <-> A e. suc B))
5		sucidg 3058	. . . . . . . . . . 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 3042	. . . . . . . . . 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 1538	. . . . . . . . . . . 12 |- (suc A = suc B -> (B e. suc A <-> B e. suc B))
14		sucidg 3058	. . . . . . . . . . . 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 3042	. . . . . . . . . . 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 1480	. . . . . . . 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 429	. . . 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 1564	. . . . 5 |- ((suc A e. V /\ -. suc B e. V) -> -. suc A = suc B)
27		sucexb 3054	. . . . 5 |- (A e. V <-> suc A e. V)
28		sucexb 3054	. . . . . 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 457	. . . 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 1564	. . . . . . 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 457	. . . . . 6 |- ((B e. V /\ -. A e. V) -> -. suc B = suc A)
35	34	ancoms 438	. . . . 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 1480	. . . 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 3050	. . . . 5 |- (-. A e. V -> suc A = A)
40		sucprc 3050	. . . . 5 |- (-. B e. V -> suc B = B)
41	39, 40	eqeqan12d 1493	. . . 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 760	. 2 |- (suc A = suc B -> A = B)
44		suceq 3040	. 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 958   e. wcel 960  Vcvv 1814  suc csuc 2956

This theorem is referenced by:  rankxpsuc 4725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-eprel 2838  df-fr 2923  df-suc 2960

Copyright terms: Public domain