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Theorem suc11g 4309
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 4306 . . . 4  |-  -.  ( B  e.  A  /\  A  e.  B )
2 sucidg 4181 . . . . . . . . . . . 12  |-  ( B  e.  W  ->  B  e.  suc  B )
3 eleq2 2117 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
42, 3syl5ibrcom 150 . . . . . . . . . . 11  |-  ( B  e.  W  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
5 elsucg 4169 . . . . . . . . . . 11  |-  ( B  e.  W  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
64, 5sylibd 142 . . . . . . . . . 10  |-  ( B  e.  W  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
76imp 119 . . . . . . . . 9  |-  ( ( B  e.  W  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
873adant1 933 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
9 sucidg 4181 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  A  e.  suc  A )
10 eleq2 2117 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
119, 10syl5ibcom 148 . . . . . . . . . . 11  |-  ( A  e.  V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
12 elsucg 4169 . . . . . . . . . . 11  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
1311, 12sylibd 142 . . . . . . . . . 10  |-  ( A  e.  V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
1413imp 119 . . . . . . . . 9  |-  ( ( A  e.  V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
15143adant2 934 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
168, 15jca 294 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( ( B  e.  A  \/  B  =  A )  /\  ( A  e.  B  \/  A  =  B
) ) )
17 eqcom 2058 . . . . . . . . 9  |-  ( B  =  A  <->  A  =  B )
1817orbi2i 689 . . . . . . . 8  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( B  e.  A  \/  A  =  B )
)
1918anbi1i 439 . . . . . . 7  |-  ( ( ( B  e.  A  \/  B  =  A
)  /\  ( A  e.  B  \/  A  =  B ) )  <->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
2016, 19sylib 131 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
21 ordir 741 . . . . . 6  |-  ( ( ( B  e.  A  /\  A  e.  B
)  \/  A  =  B )  <->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
2220, 21sylibr 141 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  suc  A  =  suc  B
)  ->  ( ( B  e.  A  /\  A  e.  B )  \/  A  =  B
) )
2322ord 653 . . . 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 1117 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
26 suceq 4167 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
2725, 26impbid1 134 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-un 2950  df-sn 3409  df-pr 3410  df-suc 4136
This theorem is referenced by:  suc11  4310  peano4  4348  frecsuclem3  6021
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