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Theorem suc11 5504
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 5411 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
2 ordn2lp 5421 . . . . . 6  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 ianor 495 . . . . . 6  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
42, 3sylib 201 . . . . 5  |-  ( Ord 
A  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
51, 4syl 17 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
65adantr 471 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  e.  B  \/  -.  B  e.  A ) )
7 eqimss 3451 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  A  C_  suc  B )
8 sucssel 5493 . . . . . 6  |-  ( A  e.  On  ->  ( suc  A  C_  suc  B  ->  A  e.  suc  B ) )
97, 8syl5 33 . . . . 5  |-  ( A  e.  On  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
10 elsuci 5467 . . . . . . 7  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
1110ord 383 . . . . . 6  |-  ( A  e.  suc  B  -> 
( -.  A  e.  B  ->  A  =  B ) )
1211com12 32 . . . . 5  |-  ( -.  A  e.  B  -> 
( A  e.  suc  B  ->  A  =  B ) )
139, 12syl9 73 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
14 eqimss2 3452 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  B  C_  suc  A )
15 sucssel 5493 . . . . . 6  |-  ( B  e.  On  ->  ( suc  B  C_  suc  A  ->  B  e.  suc  A ) )
1614, 15syl5 33 . . . . 5  |-  ( B  e.  On  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
17 elsuci 5467 . . . . . . . 8  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
1817ord 383 . . . . . . 7  |-  ( B  e.  suc  A  -> 
( -.  B  e.  A  ->  B  =  A ) )
1918com12 32 . . . . . 6  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  B  =  A ) )
20 eqcom 2458 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 234 . . . . 5  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  A  =  B ) )
2216, 21syl9 73 . . . 4  |-  ( B  e.  On  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2313, 22jaao 516 . . 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 5466 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
2624, 25impbid1 208 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1447    e. wcel 1890    C_ wss 3371   Ord word 5400   Oncon0 5401   suc csuc 5403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-br 4374  df-opab 4433  df-tr 4469  df-eprel 4722  df-po 4732  df-so 4733  df-fr 4770  df-we 4772  df-ord 5404  df-on 5405  df-suc 5407
This theorem is referenced by:  peano4  6702  limenpsi  7733  fin1a2lem2  8817  bnj168  29543  sltval2  30548  sltsolem1  30562  onsuct0  31106
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