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

Theorem suc11reg 7530
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 7527 . . . . 5  |-  -.  ( A  e.  B  /\  B  e.  A )
2 ianor 475 . . . . 5  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
31, 2mpbi 200 . . . 4  |-  ( -.  A  e.  B  \/  -.  B  e.  A
)
4 sucidg 4619 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  suc  A )
5 eleq2 2465 . . . . . . . . . . 11  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
64, 5syl5ibcom 212 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
7 elsucg 4608 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
86, 7sylibd 206 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
98imp 419 . . . . . . . 8  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
109ord 367 . . . . . . 7  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  A  e.  B  ->  A  =  B ) )
1110ex 424 . . . . . 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 4619 . . . . . . . . . . . 12  |-  ( B  e.  _V  ->  B  e.  suc  B )
14 eleq2 2465 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
1513, 14syl5ibrcom 214 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
16 elsucg 4608 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
1715, 16sylibd 206 . . . . . . . . . 10  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
1817imp 419 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
1918ord 367 . . . . . . . 8  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  B  =  A ) )
20 eqcom 2406 . . . . . . . 8  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 218 . . . . . . 7  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  A  =  B ) )
2221ex 424 . . . . . 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 496 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( -.  A  e.  B  \/  -.  B  e.  A )  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
253, 24mpi 17 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
26 sucexb 4748 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
27 sucexb 4748 . . . . . 6  |-  ( B  e.  _V  <->  suc  B  e. 
_V )
2827notbii 288 . . . . 5  |-  ( -.  B  e.  _V  <->  -.  suc  B  e.  _V )
29 nelneq 2502 . . . . 5  |-  ( ( suc  A  e.  _V  /\ 
-.  suc  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3026, 28, 29syl2anb 466 . . . 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 2406 . . . 4  |-  ( suc 
A  =  suc  B  <->  suc 
B  =  suc  A
)
3326notbii 288 . . . . . . 7  |-  ( -.  A  e.  _V  <->  -.  suc  A  e.  _V )
34 nelneq 2502 . . . . . . 7  |-  ( ( suc  B  e.  _V  /\ 
-.  suc  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3527, 33, 34syl2anb 466 . . . . . 6  |-  ( ( B  e.  _V  /\  -.  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3635ancoms 440 . . . . 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 209 . . 3  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
39 sucprc 4616 . . . . 5  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
40 sucprc 4616 . . . . 5  |-  ( -.  B  e.  _V  ->  suc 
B  =  B )
4139, 40eqeqan12d 2419 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
4241biimpd 199 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
4325, 31, 38, 424cases 916 . 2  |-  ( suc 
A  =  suc  B  ->  A  =  B )
44 suceq 4606 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
4543, 44impbii 181 1  |-  ( 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 1649    e. wcel 1721   _Vcvv 2916   suc csuc 4543
This theorem is referenced by:  rankxpsuc  7762  bnj551  28816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660  ax-reg 7516
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-eprel 4454  df-fr 4501  df-suc 4547
  Copyright terms: Public domain W3C validator