Theorem suc11 6470

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 6370	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordn2lp 6380	. . . . 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 485	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐵 ∈ 𝐴))
6		eqimss 3995	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7		sucssel 6458	. . . . . 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 6430	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 877	. . . . . 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 3996	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14		sucssel 6458	. . . . . 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 6430	. . . . . . . 8 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
17	16	ord 877	. . . . . . 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 6429	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
25	23, 24	impbid1 228	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  Ord word 6359  Oncon0 6360  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-suc 6366

This theorem is referenced by:  peano4  7885  limenpsi  9136  fin1a2lem2  10380  ltsval2  27820  ltssolem1  27839  nosepnelem  27843  nolt02o  27859  bnj168  35119  onsuct0  36972  1oequni2o  38034  onsucf1lem  44016  onsucf1o  44019