ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  suc11g Structured version   Unicode version

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  V  W  suc  suc

Proof of Theorem suc11g
StepHypRef Expression
1 en2lp 4232 . . . 4
2 sucidg 4119 . . . . . . . . . . . 12  W  suc
3 eleq2 2098 . . . . . . . . . . . 12  suc  suc  suc  suc
42, 3syl5ibrcom 146 . . . . . . . . . . 11  W  suc  suc  suc
5 elsucg 4107 . . . . . . . . . . 11  W  suc
64, 5sylibd 138 . . . . . . . . . 10  W  suc  suc
76imp 115 . . . . . . . . 9  W  suc  suc
873adant1 921 . . . . . . . 8  V  W  suc  suc
9 sucidg 4119 . . . . . . . . . . . 12  V  suc
10 eleq2 2098 . . . . . . . . . . . 12  suc  suc  suc  suc
119, 10syl5ibcom 144 . . . . . . . . . . 11  V  suc  suc  suc
12 elsucg 4107 . . . . . . . . . . 11  V  suc
1311, 12sylibd 138 . . . . . . . . . 10  V  suc  suc
1413imp 115 . . . . . . . . 9  V  suc  suc
15143adant2 922 . . . . . . . 8  V  W  suc  suc
168, 15jca 290 . . . . . . 7  V  W  suc  suc
17 eqcom 2039 . . . . . . . . 9
1817orbi2i 678 . . . . . . . 8
1918anbi1i 431 . . . . . . 7
2016, 19sylib 127 . . . . . 6  V  W  suc  suc
21 ordir 729 . . . . . 6
2220, 21sylibr 137 . . . . 5  V  W  suc  suc
2322ord 642 . . . 4  V  W  suc  suc
241, 23mpi 15 . . 3  V  W  suc  suc
25243expia 1105 . 2  V  W  suc  suc
26 suceq 4105 . 2  suc  suc
2725, 26impbid1 130 1  V  W  suc  suc
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
  Copyright terms: Public domain W3C validator