Theorem suc11reg 9535

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 9522	. . . . 5 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		ianor 990	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
3	1, 2	mpbi 232	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴)
4		sucidg 6397	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5		eleq2 2830	. . . . . . . . . . 11 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
6	4, 5	syl5ibcom 247	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
7		elsucg 6384	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8	6, 7	sylibd 241	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
9	8	imp 408	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 871	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	ex 414	. . . . . 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 6397	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14		eleq2 2830	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
15	13, 14	syl5ibrcom 249	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsucg 6384	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
17	15, 16	sylibd 241	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
18	17	imp 408	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
19	18	ord 871	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
20		eqcom 2748	. . . . . . . 8 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	19, 20	imbitrdi 253	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
22	21	ex 414	. . . . . 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 963	. . . 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 7751	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27		sucexb 7751	. . . . . 6 ⊢ (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
28	27	notbii 322	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ ¬ suc 𝐵 ∈ V)
29		nelneq 2865	. . . . 5 ⊢ ((suc 𝐴 ∈ V ∧ ¬ suc 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
30	26, 28, 29	syl2anb 605	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → ¬ suc 𝐴 = suc 𝐵)
31	30	pm2.21d 121	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
32		eqcom 2748	. . . 4 ⊢ (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
33	26	notbii 322	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ ¬ suc 𝐴 ∈ V)
34		nelneq 2865	. . . . . . 7 ⊢ ((suc 𝐵 ∈ V ∧ ¬ suc 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
35	27, 33, 34	syl2anb 605	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
36	35	ancoms 460	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
37	36	pm2.21d 121	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐵 = suc 𝐴 → 𝐴 = 𝐵))
38	32, 37	biimtrid 244	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
39		sucprc 6392	. . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
40		sucprc 6392	. . . . 5 ⊢ (¬ 𝐵 ∈ V → suc 𝐵 = 𝐵)
41	39, 40	eqeqan12d 2755	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
42	41	biimpd 231	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
43	25, 31, 38, 42	4cases 1047	. 2 ⊢ (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)
44		suceq 6382	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
45	43, 44	impbii 211	1 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  Vcvv 3433  suc csuc 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365  ax-un 7682  ax-reg 9501

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-eprel 5521  df-fr 5574  df-suc 6320

This theorem is referenced by:  rankxpsuc  9801  unidifsnel  32627  unidifsnne  32628  bnj551  34940  mopre  38853  suceqsneq  38866  sucmapleftuniq  38872  presuc  38880  clsk1indlem1  44504