Theorem suc11g 4235

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 4232	. . . 4 ⊢ ¬ (B ∈ A ∧ A ∈ B)
2		sucidg 4119	. . . . . . . . . . . 12 ⊢ (B ∈ 𝑊 → B ∈ suc B)
3		eleq2 2098	. . . . . . . . . . . 12 ⊢ (suc A = suc B → (B ∈ suc A ↔ B ∈ suc B))
4	2, 3	syl5ibrcom 146	. . . . . . . . . . 11 ⊢ (B ∈ 𝑊 → (suc A = suc B → B ∈ suc A))
5		elsucg 4107	. . . . . . . . . . 11 ⊢ (B ∈ 𝑊 → (B ∈ suc A ↔ (B ∈ A ∨ B = A)))
6	4, 5	sylibd 138	. . . . . . . . . 10 ⊢ (B ∈ 𝑊 → (suc A = suc B → (B ∈ A ∨ B = A)))
7	6	imp 115	. . . . . . . . 9 ⊢ ((B ∈ 𝑊 ∧ suc A = suc B) → (B ∈ A ∨ B = A))
8	7	3adant1 921	. . . . . . . 8 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ suc A = suc B) → (B ∈ A ∨ B = A))
9		sucidg 4119	. . . . . . . . . . . 12 ⊢ (A ∈ 𝑉 → A ∈ suc A)
10		eleq2 2098	. . . . . . . . . . . 12 ⊢ (suc A = suc B → (A ∈ suc A ↔ A ∈ suc B))
11	9, 10	syl5ibcom 144	. . . . . . . . . . 11 ⊢ (A ∈ 𝑉 → (suc A = suc B → A ∈ suc B))
12		elsucg 4107	. . . . . . . . . . 11 ⊢ (A ∈ 𝑉 → (A ∈ suc B ↔ (A ∈ B ∨ A = B)))
13	11, 12	sylibd 138	. . . . . . . . . 10 ⊢ (A ∈ 𝑉 → (suc A = suc B → (A ∈ B ∨ A = B)))
14	13	imp 115	. . . . . . . . 9 ⊢ ((A ∈ 𝑉 ∧ suc A = suc B) → (A ∈ B ∨ A = B))
15	14	3adant2 922	. . . . . . . 8 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ suc A = suc B) → (A ∈ B ∨ A = B))
16	8, 15	jca 290	. . . . . . 7 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ suc A = suc B) → ((B ∈ A ∨ B = A) ∧ (A ∈ B ∨ A = B)))
17		eqcom 2039	. . . . . . . . 9 ⊢ (B = A ↔ A = B)
18	17	orbi2i 678	. . . . . . . 8 ⊢ ((B ∈ A ∨ B = A) ↔ (B ∈ A ∨ A = B))
19	18	anbi1i 431	. . . . . . 7 ⊢ (((B ∈ A ∨ B = A) ∧ (A ∈ B ∨ A = B)) ↔ ((B ∈ A ∨ A = B) ∧ (A ∈ B ∨ A = B)))
20	16, 19	sylib 127	. . . . . 6 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ suc A = suc B) → ((B ∈ A ∨ A = B) ∧ (A ∈ B ∨ A = B)))
21		ordir 729	. . . . . 6 ⊢ (((B ∈ A ∧ A ∈ B) ∨ A = B) ↔ ((B ∈ A ∨ A = B) ∧ (A ∈ B ∨ A = B)))
22	20, 21	sylibr 137	. . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ suc A = suc B) → ((B ∈ A ∧ A ∈ B) ∨ A = B))
23	22	ord 642	. . . 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 1105	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (suc A = suc B → A = B))
26		suceq 4105	. 2 ⊢ (A = B → suc A = suc B)
27	25, 26	impbid1 130	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (suc A = suc B ↔ A = B))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-sn 3373  df-pr 3374  df-suc 4074

This theorem is referenced by:  suc11  4236  peano4  4263  frecsuclem3  5929