Theorem suc11reg 9588

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. (Contributed by NM, 25-Oct-2003.)

Assertion

Ref	Expression
suc11reg	⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)

Proof of Theorem suc11reg

Step	Hyp	Ref	Expression
1		en2lp 9575	. . . . 5 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		ianor 997	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
3	1, 2	mpbi 233	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴)
4		sucidg 6445	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5		eleq2 2858	. . . . . . . . . . 11 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
6	4, 5	syl5ibcom 248	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
7		elsucg 6432	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8	6, 7	sylibd 242	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
9	8	imp 411	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 877	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	ex 417	. . . . . 6 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵)))
12	11	com23 87	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ 𝐵 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
13		sucidg 6445	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14		eleq2 2858	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
15	13, 14	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsucg 6432	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
17	15, 16	sylibd 242	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
18	17	imp 411	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
19	18	ord 877	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
20		eqcom 2776	. . . . . . . 8 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	19, 20	imbitrdi 254	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
22	21	ex 417	. . . . . 6 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵)))
23	22	com23 87	. . . . 5 ⊢ (𝐵 ∈ V → (¬ 𝐵 ∈ 𝐴 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
24	12, 23	jaao 969	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
25	3, 24	mpi 21	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
26		sucexb 7803	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27		sucexb 7803	. . . . . 6 ⊢ (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
28	27	notbii 323	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29		nelneq 2893	. . . . 5 ⊢ ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
30	26, 28, 29	syl2anb 609	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
31	30	pm2.21d 122	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
32		eqcom 2776	. . . 4 ⊢ (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
33	26	notbii 323	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34		nelneq 2893	. . . . . . 7 ⊢ ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
35	27, 33, 34	syl2anb 609	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
36	35	ancoms 463	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
37	36	pm2.21d 122	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴 → 𝐴 = 𝐵))
38	32, 37	biimtrid 245	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
39		sucprc 6440	. . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
40		sucprc 6440	. . . . 5 ⊢ (¬ 𝐵 ∈ V → suc 𝐵 = 𝐵)
41	39, 40	eqeqan12d 2783	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
42	41	biimpd 232	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
43	25, 31, 38, 42	4cases 1054	. 2 ⊢ (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)
44		suceq 6430	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
45	43, 44	impbii 212	1 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  Vcvv 3463  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405  ax-un 7733  ax-reg 9554

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-eprel 5562  df-fr 5615  df-suc 6367

This theorem is referenced by:  rankxpsuc  9854  unidifsnel  32822  unidifsnne  32823  bnj551  35076  mopre  39044  suceqsneq  39057  sucmapleftuniq  39063  presuc  39071  clsk1indlem1  44697