Theorem suc11reg 8876

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 8864	. . . . 5 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		ianor 964	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
3	1, 2	mpbi 222	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴)
4		sucidg 6107	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5		eleq2 2854	. . . . . . . . . . 11 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
6	4, 5	syl5ibcom 237	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
7		elsucg 6096	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8	6, 7	sylibd 231	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
9	8	imp 398	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 850	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	ex 405	. . . . . 6 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵)))
12	11	com23 86	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ 𝐵 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
13		sucidg 6107	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14		eleq2 2854	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
15	13, 14	syl5ibrcom 239	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsucg 6096	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
17	15, 16	sylibd 231	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
18	17	imp 398	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
19	18	ord 850	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
20		eqcom 2785	. . . . . . . 8 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	19, 20	syl6ib 243	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
22	21	ex 405	. . . . . 6 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵)))
23	22	com23 86	. . . . 5 ⊢ (𝐵 ∈ V → (¬ 𝐵 ∈ 𝐴 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
24	12, 23	jaao 937	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
25	3, 24	mpi 20	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
26		sucexb 7340	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27		sucexb 7340	. . . . . 6 ⊢ (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
28	27	notbii 312	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29		nelneq 2890	. . . . 5 ⊢ ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
30	26, 28, 29	syl2anb 588	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
31	30	pm2.21d 119	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
32		eqcom 2785	. . . 4 ⊢ (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
33	26	notbii 312	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34		nelneq 2890	. . . . . . 7 ⊢ ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
35	27, 33, 34	syl2anb 588	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
36	35	ancoms 451	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
37	36	pm2.21d 119	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴 → 𝐴 = 𝐵))
38	32, 37	syl5bi 234	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
39		sucprc 6104	. . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
40		sucprc 6104	. . . . 5 ⊢ (¬ 𝐵 ∈ V → suc 𝐵 = 𝐵)
41	39, 40	eqeqan12d 2794	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
42	41	biimpd 221	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
43	25, 31, 38, 42	4cases 1021	. 2 ⊢ (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)
44		suceq 6094	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
45	43, 44	impbii 201	1 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833   = wceq 1507   ∈ wcel 2050  Vcvv 3415  suc csuc 6031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279  ax-reg 8851

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-eprel 5317  df-fr 5366  df-suc 6035

This theorem is referenced by:  rankxpsuc  9105  bnj551  31667  clsk1indlem1  39764