Theorem suc11 5748

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 5650	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordn2lp 5660	. . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3		pm3.13 521	. . . . 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 3620	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐴 ⊆ suc 𝐵)
7		sucssel 5736	. . . . . 6 ⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ∈ suc 𝐵))
8	6, 7	syl5 33	. . . . 5 ⊢ (𝐴 ∈ On → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
9		elsuci 5708	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
10	9	ord 391	. . . . . 6 ⊢ (𝐴 ∈ suc 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝐴 = 𝐵))
11	10	com12 32	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 = 𝐵))
12	8, 11	syl9 75	. . . 4 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 𝐵 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
13		eqimss2 3621	. . . . . 6 ⊢ (suc 𝐴 = suc 𝐵 → suc 𝐵 ⊆ suc 𝐴)
14		sucssel 5736	. . . . . 6 ⊢ (𝐵 ∈ On → (suc 𝐵 ⊆ suc 𝐴 → 𝐵 ∈ suc 𝐴))
15	13, 14	syl5 33	. . . . 5 ⊢ (𝐵 ∈ On → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
16		elsuci 5708	. . . . . . . 8 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
17	16	ord 391	. . . . . . 7 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐵 = 𝐴))
18		eqcom 2617	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
19	17, 18	syl6ib 240	. . . . . 6 ⊢ (𝐵 ∈ suc 𝐴 → (¬ 𝐵 ∈ 𝐴 → 𝐴 = 𝐵))
20	19	com12 32	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐴 = 𝐵))
21	15, 20	syl9 75	. . . 4 ⊢ (𝐵 ∈ On → (¬ 𝐵 ∈ 𝐴 → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵)))
22	12, 21	jaao 530	. . 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 5707	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
25	23, 24	impbid1 214	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639  Oncon0 5640  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646

This theorem is referenced by:  peano4  6980  limenpsi  8020  fin1a2lem2  9106  bnj168  30052  sltval2  31053  sltsolem1  31067  onsuct0  31610