Theorem suc11 2341

Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194.

Assertion

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

Proof of Theorem suc11

Step	Hyp	Ref	Expression
1		eloni 2209	. . . . 5 ⊢ (A ∈ On → Ord A)
2		ordn2lp 2219	. . . . . 6 ⊢ (Ord A → ¬ (A ∈ B ∧ B ∈ A))
3		ianor 253	. . . . . 6 ⊢ (¬ (A ∈ B ∧ B ∈ A) ↔ (¬ A ∈ B ∨ ¬ B ∈ A))
4	2, 3	sylib 173	. . . . 5 ⊢ (Ord A → (¬ A ∈ B ∨ ¬ B ∈ A))
5	1, 4	syl 12	. . . 4 ⊢ (A ∈ On → (¬ A ∈ B ∨ ¬ B ∈ A))
6	5	adantr 306	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → (¬ A ∈ B ∨ ¬ B ∈ A))
7		sucssel 2321	. . . . . 6 ⊢ (A ∈ On → (suc A ⊆ suc B → A ∈ suc B))
8		eqimss 1548	. . . . . 6 ⊢ (suc A = suc B → suc A ⊆ suc B)
9	7, 8	syl5 22	. . . . 5 ⊢ (A ∈ On → (suc A = suc B → A ∈ suc B))
10		elsuci 2289	. . . . . . 7 ⊢ (A ∈ suc B → (A ∈ B ∨ A = B))
11	10	ord 202	. . . . . 6 ⊢ (A ∈ suc B → (¬ A ∈ B → A = B))
12	11	com12 13	. . . . 5 ⊢ (¬ A ∈ B → (A ∈ suc B → A = B))
13	9, 12	syl9 55	. . . 4 ⊢ (A ∈ On → (¬ A ∈ B → (suc A = suc B → A = B)))
14		sucssel 2321	. . . . . 6 ⊢ (B ∈ On → (suc B ⊆ suc A → B ∈ suc A))
15		eqimss2 1549	. . . . . 6 ⊢ (suc A = suc B → suc B ⊆ suc A)
16	14, 15	syl5 22	. . . . 5 ⊢ (B ∈ On → (suc A = suc B → B ∈ suc A))
17		elsuci 2289	. . . . . . . 8 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
18	17	ord 202	. . . . . . 7 ⊢ (B ∈ suc A → (¬ B ∈ A → B = A))
19	18	com12 13	. . . . . 6 ⊢ (¬ B ∈ A → (B ∈ suc A → B = A))
20		cleqcom 1103	. . . . . 6 ⊢ (B = A ↔ A = B)
21	19, 20	syl6ib 185	. . . . 5 ⊢ (¬ B ∈ A → (B ∈ suc A → A = B))
22	16, 21	syl9 55	. . . 4 ⊢ (B ∈ On → (¬ B ∈ A → (suc A = suc B → A = B)))
23	13, 22	jaao 330	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → ((¬ A ∈ B ∨ ¬ B ∈ A) → (suc A = suc B → A = B)))
24	6, 23	mpd 46	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (suc A = suc B → A = B))
25		suceq 2288	. . 3 ⊢ (A = B → suc A = suc B)
26	25	a1i 7	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A = B → suc A = suc B))
27	24, 26	impbid 397	1 ⊢ ((A ∈ On ∧ B ∈ On) → (suc A = suc B ↔ A = B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  peano4 2393  limenpsi 3400

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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