Theorem suc11 6354

Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
suc11	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem suc11

Step	Hyp	Ref	Expression
1		eloni 6261	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordn2lp 6271	. . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm3.13 991	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
5	4	adantr 480	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
6		eqimss 3973	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7		sucssel 6343	. . . . . 6 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ∈ suc 𝐵))
8	6, 7	syl5 34	. . . . 5 ⊢ (𝐴 ∈ On → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
9		elsuci 6317	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 860	. . . . . 6 ⊢ (𝐴 ∈ suc 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	com12 32	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 = 𝐵))
12	8, 11	syl9 77	. . . 4 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 𝐵 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
13		eqimss2 3974	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14		sucssel 6343	. . . . . 6 ⊢ (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴 → 𝐵 ∈ suc 𝐴))
15	13, 14	syl5 34	. . . . 5 ⊢ (𝐵 ∈ On → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsuci 6317	. . . . . . . 8 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
17	16	ord 860	. . . . . . 7 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
18		eqcom 2745	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
19	17, 18	syl6ib 250	. . . . . 6 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
20	19	com12 32	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐴 = 𝐵))
21	15, 20	syl9 77	. . . 4 ⊢ (𝐵 ∈ On → (¬ 𝐵 ∈ 𝐴 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
22	12, 21	jaao 951	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
23	5, 22	mpd 15	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
24		suceq 6316	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
25	23, 24	impbid1 224	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  Ord word 6250  Oncon0 6251  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  peano4  7713  limenpsi  8888  fin1a2lem2  10088  bnj168  32609  sltval2  33786  sltsolem1  33805  nosepnelem  33809  nolt02o  33825  onsuct0  34557  1oequni2o  35466