Theorem suc11g 4581

Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)

Assertion

Ref	Expression
suc11g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem suc11g

Step	Hyp	Ref	Expression
1		en2lp 4578	. . . 4 ⊢ ¬ (𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)
2		sucidg 4441	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ suc 𝐵)
3		eleq2 2253	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	syl5ibrcom 157	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑊 → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
5		elsucg 4429	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
6	4, 5	sylibd 149	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
7	6	imp 124	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
8	7	3adant1 1017	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
9		sucidg 4441	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
10		eleq2 2253	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
11	9, 10	syl5ibcom 155	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
12		elsucg 4429	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
13	11, 12	sylibd 149	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
14	13	imp 124	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
15	14	3adant2 1018	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
16	8, 15	jca 306	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
17		eqcom 2191	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
18	17	orbi2i 763	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ↔ (𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
19	18	anbi1i 458	. . . . . . 7 ⊢ (((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) ↔ ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
20	16, 19	sylib 122	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
21		ordir 818	. . . . . 6 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
22	20, 21	sylibr 134	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) ∨ 𝐴 = 𝐵))
23	22	ord 725	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 𝐵))
24	1, 23	mpi 15	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
25	24	3expia 1207	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
26		suceq 4427	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
27	25, 26	impbid1 142	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  suc csuc 4390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4561

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2758  df-dif 3151  df-un 3153  df-sn 3620  df-pr 3621  df-suc 4396

This theorem is referenced by:  suc11  4582  peano4  4621  frecsuclem  6446