Theorem suc11 6479

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 6379	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordn2lp 6389	. . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm3.13 1010	. . . . 5 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
4	1, 2, 3	3syl 19	. . . 4 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
5	4	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
6		eqimss 3971	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7		sucssel 6467	. . . . . 6 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ∈ suc 𝐵))
8	6, 7	syl5 35	. . . . 5 ⊢ (𝐴 ∈ On → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
9		elsuci 6439	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 878	. . . . . 6 ⊢ (𝐴 ∈ suc 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	com12 33	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 = 𝐵))
12	8, 11	syl9 78	. . . 4 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 𝐵 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
13		eqimss2 3972	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14		sucssel 6467	. . . . . 6 ⊢ (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴 → 𝐵 ∈ suc 𝐴))
15	13, 14	syl5 35	. . . . 5 ⊢ (𝐵 ∈ On → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsuci 6439	. . . . . . . 8 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
17	16	ord 878	. . . . . . 7 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
18		eqcom 2770	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
19	17, 18	imbitrdi 254	. . . . . 6 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
20	19	com12 33	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐴 = 𝐵))
21	15, 20	syl9 78	. . . 4 ⊢ (𝐵 ∈ On → (¬ 𝐵 ∈ 𝐴 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
22	12, 21	jaao 969	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
23	5, 22	mpd 16	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
24		suceq 6438	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
25	23, 24	impbid1 228	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2146   ⊆ wss 3900  Ord word 6368  Oncon0 6369  suc csuc 6371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6372  df-on 6373  df-suc 6375

This theorem is used by:  peano4  7914  limenpsi  9185  fin1a2lem2  10492  ltsval2  28028  ltssolem1  28047  nosepnelem  28051  nolt02o  28067  unidifsnel  33188  unidifsnne  33189  bnj168  35417  onsuct0  37359  1oequni2o  38434  onsucf1lem  44414  onsucf1o  44417