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Theorem suc11g 4217
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 BA = B))

Proof of Theorem suc11g
StepHypRef Expression
1 en2lp 4214 . . . 4 ¬ (B A A B)
2 sucidg 4100 . . . . . . . . . . . 12 (B 𝑊B suc B)
3 eleq2 2084 . . . . . . . . . . . 12 (suc A = suc B → (B suc AB suc B))
42, 3syl5ibrcom 146 . . . . . . . . . . 11 (B 𝑊 → (suc A = suc BB suc A))
5 elsucg 4088 . . . . . . . . . . 11 (B 𝑊 → (B suc A ↔ (B A B = A)))
64, 5sylibd 138 . . . . . . . . . 10 (B 𝑊 → (suc A = suc B → (B A B = A)))
76imp 115 . . . . . . . . 9 ((B 𝑊 suc A = suc B) → (B A B = A))
873adant1 910 . . . . . . . 8 ((A 𝑉 B 𝑊 suc A = suc B) → (B A B = A))
9 sucidg 4100 . . . . . . . . . . . 12 (A 𝑉A suc A)
10 eleq2 2084 . . . . . . . . . . . 12 (suc A = suc B → (A suc AA suc B))
119, 10syl5ibcom 144 . . . . . . . . . . 11 (A 𝑉 → (suc A = suc BA suc B))
12 elsucg 4088 . . . . . . . . . . 11 (A 𝑉 → (A suc B ↔ (A B A = B)))
1311, 12sylibd 138 . . . . . . . . . 10 (A 𝑉 → (suc A = suc B → (A B A = B)))
1413imp 115 . . . . . . . . 9 ((A 𝑉 suc A = suc B) → (A B A = B))
15143adant2 911 . . . . . . . 8 ((A 𝑉 B 𝑊 suc A = suc B) → (A B A = B))
168, 15jca 290 . . . . . . 7 ((A 𝑉 B 𝑊 suc A = suc B) → ((B A B = A) (A B A = B)))
17 eqcom 2025 . . . . . . . . 9 (B = AA = B)
1817orbi2i 666 . . . . . . . 8 ((B A B = A) ↔ (B A A = B))
1918anbi1i 434 . . . . . . 7 (((B A B = A) (A B A = B)) ↔ ((B A A = B) (A B A = B)))
2016, 19sylib 127 . . . . . 6 ((A 𝑉 B 𝑊 suc A = suc B) → ((B A A = B) (A B A = B)))
21 ordir 718 . . . . . 6 (((B A A B) A = B) ↔ ((B A A = B) (A B A = B)))
2220, 21sylibr 137 . . . . 5 ((A 𝑉 B 𝑊 suc A = suc B) → ((B A A B) A = B))
2322ord 630 . . . 4 ((A 𝑉 B 𝑊 suc A = suc B) → (¬ (B A A B) → A = B))
241, 23mpi 15 . . 3 ((A 𝑉 B 𝑊 suc A = suc B) → A = B)
25243expia 1092 . 2 ((A 𝑉 B 𝑊) → (suc A = suc BA = B))
26 suceq 4086 . 2 (A = B → suc A = suc B)
2725, 26impbid1 130 1 ((A 𝑉 B 𝑊) → (suc A = suc BA = B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616   w3a 873   = wceq 1228   wcel 1375  suc csuc 4049
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-setind 4202
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-v 2536  df-dif 2896  df-un 2898  df-sn 3355  df-pr 3356  df-suc 4055
This theorem is referenced by:  suc11  4218  peano4  4245  frecsuclem3  5901
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