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Theorem suc11g 4251
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 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Proof of Theorem suc11g
StepHypRef Expression
1 en2lp 4248 . . . 4 ¬ (𝐵𝐴𝐴𝐵)
2 sucidg 4125 . . . . . . . . . . . 12 (𝐵𝑊𝐵 ∈ suc 𝐵)
3 eleq2 2101 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
42, 3syl5ibrcom 146 . . . . . . . . . . 11 (𝐵𝑊 → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
5 elsucg 4113 . . . . . . . . . . 11 (𝐵𝑊 → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
64, 5sylibd 138 . . . . . . . . . 10 (𝐵𝑊 → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
76imp 115 . . . . . . . . 9 ((𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
873adant1 922 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
9 sucidg 4125 . . . . . . . . . . . 12 (𝐴𝑉𝐴 ∈ suc 𝐴)
10 eleq2 2101 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
119, 10syl5ibcom 144 . . . . . . . . . . 11 (𝐴𝑉 → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
12 elsucg 4113 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
1311, 12sylibd 138 . . . . . . . . . 10 (𝐴𝑉 → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
1413imp 115 . . . . . . . . 9 ((𝐴𝑉 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
15143adant2 923 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
168, 15jca 290 . . . . . . 7 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)))
17 eqcom 2042 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
1817orbi2i 679 . . . . . . . 8 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
1918anbi1i 431 . . . . . . 7 (((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2016, 19sylib 127 . . . . . 6 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
21 ordir 730 . . . . . 6 (((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2220, 21sylibr 137 . . . . 5 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵))
2322ord 643 . . . 4 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵𝐴𝐴𝐵) → 𝐴 = 𝐵))
241, 23mpi 15 . . 3 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
25243expia 1106 . 2 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
26 suceq 4111 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2725, 26impbid1 130 1 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  w3a 885   = wceq 1243  wcel 1393  suc csuc 4074
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4232
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-dif 2917  df-un 2919  df-sn 3378  df-pr 3379  df-suc 4080
This theorem is referenced by:  suc11  4252  peano4  4283  frecsuclem3  5953
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