Theorem suc11 6419

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 6320	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordn2lp 6330	. . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm3.13 1002	. . . . 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 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
6		eqimss 3973	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7		sucssel 6407	. . . . . 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 6379	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 870	. . . . . 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 6407	. . . . . 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 6379	. . . . . . . 8 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
17	16	ord 870	. . . . . . 7 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
18		eqcom 2746	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
19	17, 18	imbitrdi 252	. . . . . 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 962	. . 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 6378	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
25	23, 24	impbid1 226	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  Ord word 6309  Oncon0 6310  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-suc 6316

This theorem is referenced by:  peano4  7832  limenpsi  9080  fin1a2lem2  10314  ltsval2  27638  ltssolem1  27657  nosepnelem  27661  nolt02o  27677  bnj168  34913  onsuct0  36669  1oequni2o  37730  onsucf1lem  43714  onsucf1o  43717