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Theorem suc11reg 8133
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 8127 . . . . 5  |-  -.  ( A  e.  B  /\  B  e.  A )
2 ianor 490 . . . . 5  |-  ( -.  ( A  e.  B  /\  B  e.  A
)  <->  ( -.  A  e.  B  \/  -.  B  e.  A )
)
31, 2mpbi 211 . . . 4  |-  ( -.  A  e.  B  \/  -.  B  e.  A
)
4 sucidg 5520 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  suc  A )
5 eleq2 2496 . . . . . . . . . . 11  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
64, 5syl5ibcom 223 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
7 elsucg 5509 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
86, 7sylibd 217 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
98imp 430 . . . . . . . 8  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
109ord 378 . . . . . . 7  |-  ( ( A  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  A  e.  B  ->  A  =  B ) )
1110ex 435 . . . . . 6  |-  ( A  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  A  e.  B  ->  A  =  B ) ) )
1211com23 81 . . . . 5  |-  ( A  e.  _V  ->  ( -.  A  e.  B  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
13 sucidg 5520 . . . . . . . . . . . 12  |-  ( B  e.  _V  ->  B  e.  suc  B )
14 eleq2 2496 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
1513, 14syl5ibrcom 225 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
16 elsucg 5509 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
1715, 16sylibd 217 . . . . . . . . . 10  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
1817imp 430 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
1918ord 378 . . . . . . . 8  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  B  =  A ) )
20 eqcom 2431 . . . . . . . 8  |-  ( B  =  A  <->  A  =  B )
2119, 20syl6ib 229 . . . . . . 7  |-  ( ( B  e.  _V  /\  suc  A  =  suc  B
)  ->  ( -.  B  e.  A  ->  A  =  B ) )
2221ex 435 . . . . . 6  |-  ( B  e.  _V  ->  ( suc  A  =  suc  B  ->  ( -.  B  e.  A  ->  A  =  B ) ) )
2322com23 81 . . . . 5  |-  ( B  e.  _V  ->  ( -.  B  e.  A  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
2412, 23jaao 511 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( -.  A  e.  B  \/  -.  B  e.  A )  ->  ( suc  A  =  suc  B  ->  A  =  B ) ) )
253, 24mpi 20 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
26 sucexb 6650 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
27 sucexb 6650 . . . . . 6  |-  ( B  e.  _V  <->  suc  B  e. 
_V )
2827notbii 297 . . . . 5  |-  ( -.  B  e.  _V  <->  -.  suc  B  e.  _V )
29 nelneq 2534 . . . . 5  |-  ( ( suc  A  e.  _V  /\ 
-.  suc  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3026, 28, 29syl2anb 481 . . . 4  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  -.  suc  A  =  suc  B )
3130pm2.21d 109 . . 3  |-  ( ( A  e.  _V  /\  -.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
32 eqcom 2431 . . . 4  |-  ( suc 
A  =  suc  B  <->  suc 
B  =  suc  A
)
3326notbii 297 . . . . . . 7  |-  ( -.  A  e.  _V  <->  -.  suc  A  e.  _V )
34 nelneq 2534 . . . . . . 7  |-  ( ( suc  B  e.  _V  /\ 
-.  suc  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3527, 33, 34syl2anb 481 . . . . . 6  |-  ( ( B  e.  _V  /\  -.  A  e.  _V )  ->  -.  suc  B  =  suc  A )
3635ancoms 454 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  -.  suc  B  =  suc  A )
3736pm2.21d 109 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  B  =  suc  A  ->  A  =  B ) )
3832, 37syl5bi 220 . . 3  |-  ( ( -.  A  e.  _V  /\  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
39 sucprc 5517 . . . . 5  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
40 sucprc 5517 . . . . 5  |-  ( -.  B  e.  _V  ->  suc 
B  =  B )
4139, 40eqeqan12d 2445 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
4241biimpd 210 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  B  e.  _V )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
4325, 31, 38, 424cases 957 . 2  |-  ( suc 
A  =  suc  B  ->  A  =  B )
44 suceq 5507 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
4543, 44impbii 190 1  |-  ( suc 
A  =  suc  B  <->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597  ax-reg 8116
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-eprel 4764  df-fr 4812  df-suc 5448
This theorem is referenced by:  rankxpsuc  8361  bnj551  29560  1oequni2o  31735
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