Theorem suc11reg 9659

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 9646	. . . . 5 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
2		ianor 984	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
3	1, 2	mpbi 230	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴)
4		sucidg 6465	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5		eleq2 2830	. . . . . . . . . . 11 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
6	4, 5	syl5ibcom 245	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
7		elsucg 6452	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8	6, 7	sylibd 239	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
9	8	imp 406	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 865	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	ex 412	. . . . . 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 6465	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
14		eleq2 2830	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
15	13, 14	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsucg 6452	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
17	15, 16	sylibd 239	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
18	17	imp 406	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
19	18	ord 865	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
20		eqcom 2744	. . . . . . . 8 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	19, 20	imbitrdi 251	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ suc 𝐴 = suc 𝐵) → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
22	21	ex 412	. . . . . 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 957	. . . 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 7824	. . . . 5 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
27		sucexb 7824	. . . . . 6 ⊢ (𝐵 ∈ V ↔ suc 𝐵 ∈ V)
28	27	notbii 320	. . . . 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 598	. . . 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 2744	. . . 4 ⊢ (suc 𝐴 = suc 𝐵 ↔ suc 𝐵 = suc 𝐴)
33	26	notbii 320	. . . . . . 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 598	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐴 ∈ V) → ¬ suc 𝐵 = suc 𝐴)
36	35	ancoms 458	. . . . 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 242	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
39		sucprc 6460	. . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
40		sucprc 6460	. . . . 5 ⊢ (¬ 𝐵 ∈ V → suc 𝐵 = 𝐵)
41	39, 40	eqeqan12d 2751	. . . 4 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
42	41	biimpd 229	. . 3 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
43	25, 31, 38, 42	4cases 1041	. 2 ⊢ (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)
44		suceq 6450	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
45	43, 44	impbii 209	1 ⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  Vcvv 3480  suc csuc 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637  df-suc 6390

This theorem is referenced by:  rankxpsuc  9922  unidifsnel  32553  unidifsnne  32554  bnj551  34756  suceqsneq  38238  clsk1indlem1  44058