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Theorem suc11g 4373
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  e.  V  /\  B  e.  W )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )

Proof of Theorem suc11g
StepHypRef Expression
1 en2lp 4370 . . . 4  |-  -.  ( B  e.  A  /\  A  e.  B )
2 sucidg 4243 . . . . . . . . . . . 12  |-  ( B  e.  W  ->  B  e.  suc  B )
3 eleq2 2151 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
42, 3syl5ibrcom 155 . . . . . . . . . . 11  |-  ( B  e.  W  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
5 elsucg 4231 . . . . . . . . . . 11  |-  ( B  e.  W  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
64, 5sylibd 147 . . . . . . . . . 10  |-  ( B  e.  W  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
76imp 122 . . . . . . . . 9  |-  ( ( B  e.  W  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
873adant1 961 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
9 sucidg 4243 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  A  e.  suc  A )
10 eleq2 2151 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
119, 10syl5ibcom 153 . . . . . . . . . . 11  |-  ( A  e.  V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
12 elsucg 4231 . . . . . . . . . . 11  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
1311, 12sylibd 147 . . . . . . . . . 10  |-  ( A  e.  V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
1413imp 122 . . . . . . . . 9  |-  ( ( A  e.  V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
15143adant2 962 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
168, 15jca 300 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( ( B  e.  A  \/  B  =  A )  /\  ( A  e.  B  \/  A  =  B
) ) )
17 eqcom 2090 . . . . . . . . 9  |-  ( B  =  A  <->  A  =  B )
1817orbi2i 714 . . . . . . . 8  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( B  e.  A  \/  A  =  B )
)
1918anbi1i 446 . . . . . . 7  |-  ( ( ( B  e.  A  \/  B  =  A
)  /\  ( A  e.  B  \/  A  =  B ) )  <->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
2016, 19sylib 120 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
21 ordir 766 . . . . . 6  |-  ( ( ( B  e.  A  /\  A  e.  B
)  \/  A  =  B )  <->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
2220, 21sylibr 132 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( ( B  e.  A  /\  A  e.  B )  \/  A  =  B
) )
2322ord 678 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( -.  ( B  e.  A  /\  A  e.  B
)  ->  A  =  B ) )
241, 23mpi 15 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  A  =  B )
25243expia 1145 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
26 suceq 4229 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
2725, 26impbid1 140 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   suc csuc 4192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 3001  df-un 3003  df-sn 3452  df-pr 3453  df-suc 4198
This theorem is referenced by:  suc11  4374  peano4  4412  frecsuclem  6171
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