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Theorem suc11g 4302
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 4299 . . . 4 ¬ (𝐵𝐴𝐴𝐵)
2 sucidg 4173 . . . . . . . . . . . 12 (𝐵𝑊𝐵 ∈ suc 𝐵)
3 eleq2 2143 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
42, 3syl5ibrcom 155 . . . . . . . . . . 11 (𝐵𝑊 → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
5 elsucg 4161 . . . . . . . . . . 11 (𝐵𝑊 → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
64, 5sylibd 147 . . . . . . . . . 10 (𝐵𝑊 → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
76imp 122 . . . . . . . . 9 ((𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
873adant1 957 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
9 sucidg 4173 . . . . . . . . . . . 12 (𝐴𝑉𝐴 ∈ suc 𝐴)
10 eleq2 2143 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
119, 10syl5ibcom 153 . . . . . . . . . . 11 (𝐴𝑉 → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
12 elsucg 4161 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
1311, 12sylibd 147 . . . . . . . . . 10 (𝐴𝑉 → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
1413imp 122 . . . . . . . . 9 ((𝐴𝑉 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
15143adant2 958 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
168, 15jca 300 . . . . . . 7 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)))
17 eqcom 2084 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
1817orbi2i 712 . . . . . . . 8 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
1918anbi1i 446 . . . . . . 7 (((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2016, 19sylib 120 . . . . . 6 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
21 ordir 764 . . . . . 6 (((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2220, 21sylibr 132 . . . . 5 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵))
2322ord 676 . . . 4 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵𝐴𝐴𝐵) → 𝐴 = 𝐵))
241, 23mpi 15 . . 3 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
25243expia 1141 . 2 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
26 suceq 4159 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
2725, 26impbid1 140 1 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  suc csuc 4122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4282
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-sn 3406  df-pr 3407  df-suc 4128
This theorem is referenced by:  suc11  4303  peano4  4340  frecsuclem  6049
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