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Theorem suc11 4677
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 4583 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
2 ordn2lp 4593 . . . . . 6  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 ianor 475 . . . . . 6  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
42, 3sylib 189 . . . . 5  |-  ( Ord 
A  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
51, 4syl 16 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
65adantr 452 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  e.  B  \/  -.  B  e.  A ) )
7 eqimss 3392 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  A  C_  suc  B )
8 sucssel 4666 . . . . . 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 4639 . . . . . . 7  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
1110ord 367 . . . . . 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 3393 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  B  C_  suc  A )
15 sucssel 4666 . . . . . 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 4639 . . . . . . . 8  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
1817ord 367 . . . . . . 7  |-  ( B  e.  suc  A  -> 
( -.  B  e.  A  ->  B  =  A ) )
1918com12 29 . . . . . 6  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  B  =  A ) )
20 eqcom 2437 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 218 . . . . 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 496 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( -.  A  e.  B  \/  -.  B  e.  A )  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
246, 23mpd 15 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
25 suceq 4638 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
2624, 25impbid1 195 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem is referenced by:  peano4  4859  limenpsi  7274  fin1a2lem2  8273  sltval2  25603  sltsolem1  25615  onsuct0  26183  bnj168  29034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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