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Theorem suc11 4495
Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
suc11  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )

Proof of Theorem suc11
StepHypRef Expression
1 eloni 4401 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
2 ordn2lp 4411 . . . . . 6  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 ianor 476 . . . . . 6  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
42, 3sylib 190 . . . . 5  |-  ( Ord 
A  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
51, 4syl 17 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
65adantr 453 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  e.  B  \/  -.  B  e.  A ) )
7 eqimss 3231 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  A  C_  suc  B )
8 sucssel 4484 . . . . . 6  |-  ( A  e.  On  ->  ( suc  A  C_  suc  B  ->  A  e.  suc  B ) )
97, 8syl5 30 . . . . 5  |-  ( A  e.  On  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
10 elsuci 4457 . . . . . . 7  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
1110ord 368 . . . . . 6  |-  ( A  e.  suc  B  -> 
( -.  A  e.  B  ->  A  =  B ) )
1211com12 29 . . . . 5  |-  ( -.  A  e.  B  -> 
( A  e.  suc  B  ->  A  =  B ) )
139, 12syl9 68 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
14 eqimss2 3232 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  B  C_  suc  A )
15 sucssel 4484 . . . . . 6  |-  ( B  e.  On  ->  ( suc  B  C_  suc  A  ->  B  e.  suc  A ) )
1614, 15syl5 30 . . . . 5  |-  ( B  e.  On  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
17 elsuci 4457 . . . . . . . 8  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
1817ord 368 . . . . . . 7  |-  ( B  e.  suc  A  -> 
( -.  B  e.  A  ->  B  =  A ) )
1918com12 29 . . . . . 6  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  B  =  A ) )
20 eqcom 2286 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 219 . . . . 5  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  A  =  B ) )
2216, 21syl9 68 . . . 4  |-  ( B  e.  On  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2313, 22jaao 497 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( -.  A  e.  B  \/  -.  B  e.  A )  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
246, 23mpd 16 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
25 suceq 4456 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
2624, 25impbid1 196 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685    C_ wss 3153   Ord word 4390   Oncon0 4391   suc csuc 4393
This theorem is referenced by:  peano4  4677  limenpsi  7031  fin1a2lem2  8022  sltval2  23710  axsltsolem1  23722  onsuct0  24287  bnj168  28025
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397
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