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Theorem suc11g 4219
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 4216 . . . 4
2 sucidg 4102 . . . . . . . . . . . 12  W  suc
3 eleq2 2083 . . . . . . . . . . . 12  suc  suc  suc  suc
42, 3syl5ibrcom 146 . . . . . . . . . . 11  W  suc  suc  suc
5 elsucg 4090 . . . . . . . . . . 11  W  suc
64, 5sylibd 138 . . . . . . . . . 10  W  suc  suc
76imp 115 . . . . . . . . 9  W  suc  suc
873adant1 910 . . . . . . . 8  V  W  suc  suc
9 sucidg 4102 . . . . . . . . . . . 12  V  suc
10 eleq2 2083 . . . . . . . . . . . 12  suc  suc  suc  suc
119, 10syl5ibcom 144 . . . . . . . . . . 11  V  suc  suc  suc
12 elsucg 4090 . . . . . . . . . . 11  V  suc
1311, 12sylibd 138 . . . . . . . . . 10  V  suc  suc
1413imp 115 . . . . . . . . 9  V  suc  suc
15143adant2 911 . . . . . . . 8  V  W  suc  suc
168, 15jca 290 . . . . . . 7  V  W  suc  suc
17 eqcom 2024 . . . . . . . . 9
1817orbi2i 666 . . . . . . . 8
1918anbi1i 434 . . . . . . 7
2016, 19sylib 127 . . . . . 6  V  W  suc  suc
21 ordir 718 . . . . . 6
2220, 21sylibr 137 . . . . 5  V  W  suc  suc
2322ord 630 . . . 4  V  W  suc  suc
241, 23mpi 15 . . 3  V  W  suc  suc
25243expia 1092 . 2  V  W  suc  suc
26 suceq 4088 . 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 616   w3a 873   wceq 1228   wcel 1374   suc csuc 4051
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-un 2899  df-sn 3356  df-pr 3357  df-suc 4057
This theorem is referenced by:  suc11  4220  peano4  4247  frecsuclem3  5906
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