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Theorem suc11reg 7320
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 7317 . . . . 5  |-  -.  ( A  e.  B  /\  B  e.  A )
2 ianor 474 . . . . 5  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
31, 2mpbi 199 . . . 4  |-  ( -.  A  e.  B  \/  -.  B  e.  A
)
4 sucidg 4470 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  suc  A )
5 eleq2 2344 . . . . . . . . . . 11  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
64, 5syl5ibcom 211 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
7 elsucg 4459 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
86, 7sylibd 205 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
98imp 418 . . . . . . . 8  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
109ord 366 . . . . . . 7  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  A  e.  B  ->  A  =  B ) )
1110ex 423 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  A  e.  B  ->  A  =  B ) ) )
1211com23 72 . . . . 5  |-  ( A  e.  _V  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
13 sucidg 4470 . . . . . . . . . . . 12  |-  ( B  e.  _V  ->  B  e.  suc  B )
14 eleq2 2344 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
1513, 14syl5ibrcom 213 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
16 elsucg 4459 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
1715, 16sylibd 205 . . . . . . . . . 10  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
1817imp 418 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
1918ord 366 . . . . . . . 8  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  B  =  A ) )
20 eqcom 2285 . . . . . . . 8  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 217 . . . . . . 7  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  A  =  B ) )
2221ex 423 . . . . . 6  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  B  e.  A  ->  A  =  B ) ) )
2322com23 72 . . . . 5  |-  ( B  e.  _V  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2412, 23jaao 495 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( -.  A  e.  B  \/  -.  B  e.  A )  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
253, 24mpi 16 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
26 sucexb 4600 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
27 sucexb 4600 . . . . . 6  |-  ( B  e.  _V  <->  suc  B  e. 
_V )
2827notbii 287 . . . . 5  |-  ( -.  B  e.  _V  <->  -.  suc  B  e.  _V )
29 nelneq 2381 . . . . 5  |-  ( ( suc  A  e.  _V  /\ 
-.  suc  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3026, 28, 29syl2anb 465 . . . 4  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3130pm2.21d 98 . . 3  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
32 eqcom 2285 . . . 4  |-  ( suc 
A  =  suc  B  <->  suc 
B  =  suc  A
)
3326notbii 287 . . . . . . 7  |-  ( -.  A  e.  _V  <->  -.  suc  A  e.  _V )
34 nelneq 2381 . . . . . . 7  |-  ( ( suc  B  e.  _V  /\ 
-.  suc  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3527, 33, 34syl2anb 465 . . . . . 6  |-  ( ( B  e.  _V  /\  -.  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3635ancoms 439 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  -.  suc  B  =  suc  A )
3736pm2.21d 98 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  B  =  suc  A  ->  A  =  B ) )
3832, 37syl5bi 208 . . 3  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
39 sucprc 4467 . . . . 5  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
40 sucprc 4467 . . . . 5  |-  ( -.  B  e.  _V  ->  suc 
B  =  B )
4139, 40eqeqan12d 2298 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
4241biimpd 198 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
4325, 31, 38, 424cases 915 . 2  |-  ( suc 
A  =  suc  B  ->  A  =  B )
44 suceq 4457 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
4543, 44impbii 180 1  |-  ( suc 
A  =  suc  B  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   suc csuc 4394
This theorem is referenced by:  rankxpsuc  7552  bnj551  28771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352  df-suc 4398
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