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Theorem suc11g 4442
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 4439 . . . 4 ¬ (𝐵𝐴𝐴𝐵)
2 sucidg 4308 . . . . . . . . . . . 12 (𝐵𝑊𝐵 ∈ suc 𝐵)
3 eleq2 2181 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
42, 3syl5ibrcom 156 . . . . . . . . . . 11 (𝐵𝑊 → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
5 elsucg 4296 . . . . . . . . . . 11 (𝐵𝑊 → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
64, 5sylibd 148 . . . . . . . . . 10 (𝐵𝑊 → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
76imp 123 . . . . . . . . 9 ((𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
873adant1 984 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
9 sucidg 4308 . . . . . . . . . . . 12 (𝐴𝑉𝐴 ∈ suc 𝐴)
10 eleq2 2181 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
119, 10syl5ibcom 154 . . . . . . . . . . 11 (𝐴𝑉 → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
12 elsucg 4296 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
1311, 12sylibd 148 . . . . . . . . . 10 (𝐴𝑉 → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
1413imp 123 . . . . . . . . 9 ((𝐴𝑉 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
15143adant2 985 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
168, 15jca 304 . . . . . . 7 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)))
17 eqcom 2119 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
1817orbi2i 736 . . . . . . . 8 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
1918anbi1i 453 . . . . . . 7 (((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2016, 19sylib 121 . . . . . 6 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
21 ordir 791 . . . . . 6 (((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2220, 21sylibr 133 . . . . 5 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵))
2322ord 698 . . . 4 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵𝐴𝐴𝐵) → 𝐴 = 𝐵))
241, 23mpi 15 . . 3 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
25243expia 1168 . 2 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
26 suceq 4294 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2725, 26impbid1 141 1 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682  w3a 947   = wceq 1316  wcel 1465  suc csuc 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-dif 3043  df-un 3045  df-sn 3503  df-pr 3504  df-suc 4263
This theorem is referenced by:  suc11  4443  peano4  4481  frecsuclem  6271
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