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Theorem suc11g 4235
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 4232 . . . 4 ¬ (B A A B)
2 sucidg 4119 . . . . . . . . . . . 12 (B 𝑊B suc B)
3 eleq2 2098 . . . . . . . . . . . 12 (suc A = suc B → (B suc AB suc B))
42, 3syl5ibrcom 146 . . . . . . . . . . 11 (B 𝑊 → (suc A = suc BB suc A))
5 elsucg 4107 . . . . . . . . . . 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 921 . . . . . . . 8 ((A 𝑉 B 𝑊 suc A = suc B) → (B A B = A))
9 sucidg 4119 . . . . . . . . . . . 12 (A 𝑉A suc A)
10 eleq2 2098 . . . . . . . . . . . 12 (suc A = suc B → (A suc AA suc B))
119, 10syl5ibcom 144 . . . . . . . . . . 11 (A 𝑉 → (suc A = suc BA suc B))
12 elsucg 4107 . . . . . . . . . . 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 922 . . . . . . . 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 2039 . . . . . . . . 9 (B = AA = B)
1817orbi2i 678 . . . . . . . 8 ((B A B = A) ↔ (B A A = B))
1918anbi1i 431 . . . . . . 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 729 . . . . . 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 642 . . . 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 1105 . 2 ((A 𝑉 B 𝑊) → (suc A = suc BA = B))
26 suceq 4105 . 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 628   w3a 884   = wceq 1242   wcel 1390  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-sn 3373  df-pr 3374  df-suc 4074
This theorem is referenced by:  suc11  4236  peano4  4263  frecsuclem3  5929
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