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Theorem suc11reg 8027
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 8021 . . . . 5  |-  -.  ( A  e.  B  /\  B  e.  A )
2 ianor 488 . . . . 5  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
31, 2mpbi 208 . . . 4  |-  ( -.  A  e.  B  \/  -.  B  e.  A
)
4 sucidg 4951 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  suc  A )
5 eleq2 2535 . . . . . . . . . . 11  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
64, 5syl5ibcom 220 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
7 elsucg 4940 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
86, 7sylibd 214 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
98imp 429 . . . . . . . 8  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
109ord 377 . . . . . . 7  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  A  e.  B  ->  A  =  B ) )
1110ex 434 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  A  e.  B  ->  A  =  B ) ) )
1211com23 78 . . . . 5  |-  ( A  e.  _V  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
13 sucidg 4951 . . . . . . . . . . . 12  |-  ( B  e.  _V  ->  B  e.  suc  B )
14 eleq2 2535 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
1513, 14syl5ibrcom 222 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
16 elsucg 4940 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
1715, 16sylibd 214 . . . . . . . . . 10  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
1817imp 429 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
1918ord 377 . . . . . . . 8  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  B  =  A ) )
20 eqcom 2471 . . . . . . . 8  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 226 . . . . . . 7  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  A  =  B ) )
2221ex 434 . . . . . 6  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  B  e.  A  ->  A  =  B ) ) )
2322com23 78 . . . . 5  |-  ( B  e.  _V  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2412, 23jaao 509 . . . 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 6617 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
27 sucexb 6617 . . . . . 6  |-  ( B  e.  _V  <->  suc  B  e. 
_V )
2827notbii 296 . . . . 5  |-  ( -.  B  e.  _V  <->  -.  suc  B  e.  _V )
29 nelneq 2579 . . . . 5  |-  ( ( suc  A  e.  _V  /\ 
-.  suc  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3026, 28, 29syl2anb 479 . . . 4  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3130pm2.21d 106 . . 3  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
32 eqcom 2471 . . . 4  |-  ( suc 
A  =  suc  B  <->  suc 
B  =  suc  A
)
3326notbii 296 . . . . . . 7  |-  ( -.  A  e.  _V  <->  -.  suc  A  e.  _V )
34 nelneq 2579 . . . . . . 7  |-  ( ( suc  B  e.  _V  /\ 
-.  suc  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3527, 33, 34syl2anb 479 . . . . . 6  |-  ( ( B  e.  _V  /\  -.  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3635ancoms 453 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  -.  suc  B  =  suc  A )
3736pm2.21d 106 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  B  =  suc  A  ->  A  =  B ) )
3832, 37syl5bi 217 . . 3  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
39 sucprc 4948 . . . . 5  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
40 sucprc 4948 . . . . 5  |-  ( -.  B  e.  _V  ->  suc 
B  =  B )
4139, 40eqeqan12d 2485 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
4241biimpd 207 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
4325, 31, 38, 424cases 942 . 2  |-  ( suc 
A  =  suc  B  ->  A  =  B )
44 suceq 4938 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
4543, 44impbii 188 1  |-  ( 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 1374    e. wcel 1762   _Vcvv 3108   suc csuc 4875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569  ax-reg 8009
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-eprel 4786  df-fr 4833  df-suc 4879
This theorem is referenced by:  rankxpsuc  8291  bnj551  32755
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