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Theorem suc11 4468
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 4374 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
2 ordn2lp 4384 . . . . . 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 3205 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  A  C_  suc  B )
8 sucssel 4457 . . . . . 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 4430 . . . . . . 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 3206 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  B  C_  suc  A )
15 sucssel 4457 . . . . . 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 4430 . . . . . . . 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 2260 . . . . . 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 4429 . 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 1619    e. wcel 1621    C_ wss 3127   Ord word 4363   Oncon0 4364   suc csuc 4366
This theorem is referenced by:  peano4  4650  limenpsi  7004  fin1a2lem2  7995  sltval2  23678  axsltsolem1  23690  onsuct0  24255  bnj168  27807
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370
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