Theorem suc11reg 9584

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 9571	. . . . 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 6444	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5		eleq2 2852	. . . . . . . . . . 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 6431	. . . . . . . . . 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 6444	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14		eleq2 2852	. . . . . . . . . . . 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 6431	. . . . . . . . . . 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 2770	. . . . . . . 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 7799	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27		sucexb 7799	. . . . . 6 ⊢ (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
28	27	notbii 323	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29		nelneq 2887	. . . . 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 2770	. . . 4 ⊢ (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
33	26	notbii 323	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34		nelneq 2887	. . . . . . 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 6439	. . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
40		sucprc 6439	. . . . 5 ⊢ (¬ 𝐵 ∈ V → suc 𝐵 = 𝐵)
41	39, 40	eqeqan12d 2777	. . . 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 1056	. 2 ⊢ (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)
44		suceq 6429	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
45	43, 44	impbii 212	1 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455  suc csuc 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732  ax-reg 9550

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-eprel 5561  df-fr 5614  df-suc 6366

This theorem is used by:  rankxpsuc  9850  unidifsnel  32890  unidifsnne  32891  bnj551  35140  mopre  39148  suceqsneq  39161  sucmapleftuniq  39167  presuc  39175  clsk1indlem1  44799