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Theorem suc11g 4401
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 4398 . . . 4 ¬ (𝐵𝐴𝐴𝐵)
2 sucidg 4267 . . . . . . . . . . . 12 (𝐵𝑊𝐵 ∈ suc 𝐵)
3 eleq2 2158 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
42, 3syl5ibrcom 156 . . . . . . . . . . 11 (𝐵𝑊 → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
5 elsucg 4255 . . . . . . . . . . 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 964 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
9 sucidg 4267 . . . . . . . . . . . 12 (𝐴𝑉𝐴 ∈ suc 𝐴)
10 eleq2 2158 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
119, 10syl5ibcom 154 . . . . . . . . . . 11 (𝐴𝑉 → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
12 elsucg 4255 . . . . . . . . . . 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 965 . . . . . . . 8 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
168, 15jca 301 . . . . . . 7 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)))
17 eqcom 2097 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
1817orbi2i 717 . . . . . . . 8 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
1918anbi1i 447 . . . . . . 7 (((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2016, 19sylib 121 . . . . . 6 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
21 ordir 769 . . . . . 6 (((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2220, 21sylibr 133 . . . . 5 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵))
2322ord 681 . . . 4 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵𝐴𝐴𝐵) → 𝐴 = 𝐵))
241, 23mpi 15 . . 3 ((𝐴𝑉𝐵𝑊 ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
25243expia 1148 . 2 ((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
26 suceq 4253 . 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 667  w3a 927   = wceq 1296  wcel 1445  suc csuc 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-dif 3015  df-un 3017  df-sn 3472  df-pr 3473  df-suc 4222
This theorem is referenced by:  suc11  4402  peano4  4440  frecsuclem  6209
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