MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suc11 Structured version   Unicode version

Theorem suc11 4827
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 4734 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
2 ordn2lp 4744 . . . . . 6  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
3 ianor 488 . . . . . 6  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
42, 3sylib 196 . . . . 5  |-  ( Ord 
A  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
51, 4syl 16 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  \/  -.  B  e.  A
) )
65adantr 465 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( -.  A  e.  B  \/  -.  B  e.  A ) )
7 eqimss 3413 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  A  C_  suc  B )
8 sucssel 4816 . . . . . 6  |-  ( A  e.  On  ->  ( suc  A  C_  suc  B  ->  A  e.  suc  B ) )
97, 8syl5 32 . . . . 5  |-  ( A  e.  On  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
10 elsuci 4790 . . . . . . 7  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
1110ord 377 . . . . . 6  |-  ( A  e.  suc  B  -> 
( -.  A  e.  B  ->  A  =  B ) )
1211com12 31 . . . . 5  |-  ( -.  A  e.  B  -> 
( A  e.  suc  B  ->  A  =  B ) )
139, 12syl9 71 . . . 4  |-  ( A  e.  On  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
14 eqimss2 3414 . . . . . 6  |-  ( suc 
A  =  suc  B  ->  suc  B  C_  suc  A )
15 sucssel 4816 . . . . . 6  |-  ( B  e.  On  ->  ( suc  B  C_  suc  A  ->  B  e.  suc  A ) )
1614, 15syl5 32 . . . . 5  |-  ( B  e.  On  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
17 elsuci 4790 . . . . . . . 8  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
1817ord 377 . . . . . . 7  |-  ( B  e.  suc  A  -> 
( -.  B  e.  A  ->  B  =  A ) )
1918com12 31 . . . . . 6  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  B  =  A ) )
20 eqcom 2445 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 226 . . . . 5  |-  ( -.  B  e.  A  -> 
( B  e.  suc  A  ->  A  =  B ) )
2216, 21syl9 71 . . . 4  |-  ( B  e.  On  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2313, 22jaao 509 . . 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 4789 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
2624, 25impbid1 203 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 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3333   Ord word 4723   Oncon0 4724   suc csuc 4726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-suc 4730
This theorem is referenced by:  peano4  6503  limenpsi  7491  fin1a2lem2  8575  sltval2  27802  sltsolem1  27814  onsuct0  28292  bnj168  31726
  Copyright terms: Public domain W3C validator