MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suc11reg Unicode version

Theorem suc11reg 7315
Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
Assertion
Ref Expression
suc11reg  |-  ( suc 
A  =  suc  B  <->  A  =  B )

Proof of Theorem suc11reg
StepHypRef Expression
1 en2lp 7312 . . . . 5  |-  -.  ( A  e.  B  /\  B  e.  A )
2 ianor 476 . . . . 5  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
31, 2mpbi 201 . . . 4  |-  ( -.  A  e.  B  \/  -.  B  e.  A
)
4 sucidg 4469 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  suc  A )
5 eleq2 2345 . . . . . . . . . . 11  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
64, 5syl5ibcom 213 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
7 elsucg 4458 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
86, 7sylibd 207 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
98imp 420 . . . . . . . 8  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
109ord 368 . . . . . . 7  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  A  e.  B  ->  A  =  B ) )
1110ex 425 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  A  e.  B  ->  A  =  B ) ) )
1211com23 74 . . . . 5  |-  ( A  e.  _V  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
13 sucidg 4469 . . . . . . . . . . . 12  |-  ( B  e.  _V  ->  B  e.  suc  B )
14 eleq2 2345 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
1513, 14syl5ibrcom 215 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
16 elsucg 4458 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
1715, 16sylibd 207 . . . . . . . . . 10  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
1817imp 420 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
1918ord 368 . . . . . . . 8  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  B  =  A ) )
20 eqcom 2286 . . . . . . . 8  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 219 . . . . . . 7  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  A  =  B ) )
2221ex 425 . . . . . 6  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  B  e.  A  ->  A  =  B ) ) )
2322com23 74 . . . . 5  |-  ( B  e.  _V  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2412, 23jaao 497 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( -.  A  e.  B  \/  -.  B  e.  A )  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
253, 24mpi 18 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
26 sucexb 4599 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
27 sucexb 4599 . . . . . 6  |-  ( B  e.  _V  <->  suc  B  e. 
_V )
2827notbii 289 . . . . 5  |-  ( -.  B  e.  _V  <->  -.  suc  B  e.  _V )
29 nelneq 2382 . . . . 5  |-  ( ( suc  A  e.  _V  /\ 
-.  suc  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3026, 28, 29syl2anb 467 . . . 4  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3130pm2.21d 100 . . 3  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
32 eqcom 2286 . . . 4  |-  ( suc 
A  =  suc  B  <->  suc 
B  =  suc  A
)
3326notbii 289 . . . . . . 7  |-  ( -.  A  e.  _V  <->  -.  suc  A  e.  _V )
34 nelneq 2382 . . . . . . 7  |-  ( ( suc  B  e.  _V  /\ 
-.  suc  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3527, 33, 34syl2anb 467 . . . . . 6  |-  ( ( B  e.  _V  /\  -.  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3635ancoms 441 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  -.  suc  B  =  suc  A )
3736pm2.21d 100 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  B  =  suc  A  ->  A  =  B ) )
3832, 37syl5bi 210 . . 3  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
39 sucprc 4466 . . . . 5  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
40 sucprc 4466 . . . . 5  |-  ( -.  B  e.  _V  ->  suc 
B  =  B )
4139, 40eqeqan12d 2299 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
4241biimpd 200 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
4325, 31, 38, 424cases 917 . 2  |-  ( suc 
A  =  suc  B  ->  A  =  B )
44 suceq 4456 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
4543, 44impbii 182 1  |-  ( 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   _Vcvv 2789   suc csuc 4393
This theorem is referenced by:  rankxpsuc  7547  bnj551  28038
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-13 1687  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  ax-un 4511  ax-reg 7301
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-eprel 4304  df-fr 4351  df-suc 4397
  Copyright terms: Public domain W3C validator