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Theorem brsuccf 25694
Description: Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brsuccf.1  |-  A  e. 
_V
brsuccf.2  |-  B  e. 
_V
Assertion
Ref Expression
brsuccf  |-  ( ASucc B  <->  B  =  suc  A )

Proof of Theorem brsuccf
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-succf 25627 . . 3  |- Succ  =  (Cup 
o.  (  _I  (x) Singleton ) )
21breqi 4178 . 2  |-  ( ASucc B  <->  A (Cup  o.  (  _I  (x) Singleton ) ) B )
3 brsuccf.1 . . . . 5  |-  A  e. 
_V
4 brsuccf.2 . . . . 5  |-  B  e. 
_V
53, 4brco 5002 . . . 4  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  E. x
( A (  _I 
(x) Singleton ) x  /\  xCup B ) )
63brtxp2 25635 . . . . . . . . 9  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
) )
7 3anass 940 . . . . . . . . . . 11  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) ) )
8 vex 2919 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
98ideq 4984 . . . . . . . . . . . . . 14  |-  ( A  _I  a  <->  A  =  a )
10 eqcom 2406 . . . . . . . . . . . . . 14  |-  ( A  =  a  <->  a  =  A )
119, 10bitri 241 . . . . . . . . . . . . 13  |-  ( A  _I  a  <->  a  =  A )
12 vex 2919 . . . . . . . . . . . . . 14  |-  b  e. 
_V
133, 12brsingle 25670 . . . . . . . . . . . . 13  |-  ( ASingleton
b  <->  b  =  { A } )
1411, 13anbi12i 679 . . . . . . . . . . . 12  |-  ( ( A  _I  a  /\  ASingleton b )  <->  ( a  =  A  /\  b  =  { A } ) )
1514anbi2i 676 . . . . . . . . . . 11  |-  ( ( x  =  <. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( x  =  <. a ,  b
>.  /\  ( a  =  A  /\  b  =  { A } ) ) )
167, 15bitri 241 . . . . . . . . . 10  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) ) )
17162exbii 1590 . . . . . . . . 9  |-  ( E. a E. b ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  E. a E. b
( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) ) )
186, 17bitri 241 . . . . . . . 8  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) ) )
1918anbi1i 677 . . . . . . 7  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  ( E. a E. b ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) )  /\  xCup B ) )
20 19.41vv 1921 . . . . . . 7  |-  ( E. a E. b ( ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) )  /\  xCup B )  <->  ( E. a E. b ( x  =  <. a ,  b
>.  /\  ( a  =  A  /\  b  =  { A } ) )  /\  xCup B
) )
2119, 20bitr4i 244 . . . . . 6  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  E. a E. b
( ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) )  /\  xCup B ) )
22 anass 631 . . . . . . 7  |-  ( ( ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) )  /\  xCup B )  <->  ( x  =  <. a ,  b
>.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
23222exbii 1590 . . . . . 6  |-  ( E. a E. b ( ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) )  /\  xCup B )  <->  E. a E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
2421, 23bitri 241 . . . . 5  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  E. a E. b
( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
2524exbii 1589 . . . 4  |-  ( E. x ( A (  _I  (x) Singleton ) x  /\  xCup B )  <->  E. x E. a E. b ( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
26 excom 1752 . . . . 5  |-  ( E. x E. a E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  E. a E. x E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
27 excom 1752 . . . . . . 7  |-  ( E. x E. b ( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  E. b E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
28 opex 4387 . . . . . . . . . 10  |-  <. a ,  b >.  e.  _V
29 breq1 4175 . . . . . . . . . . . 12  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  <.
a ,  b >.Cup B ) )
308, 12, 4brcup 25692 . . . . . . . . . . . 12  |-  ( <.
a ,  b >.Cup B 
<->  B  =  ( a  u.  b ) )
3129, 30syl6bb 253 . . . . . . . . . . 11  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  B  =  ( a  u.  b ) ) )
3231anbi2d 685 . . . . . . . . . 10  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( a  =  A  /\  b  =  { A } )  /\  xCup B )  <->  ( (
a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) ) )
3328, 32ceqsexv 2951 . . . . . . . . 9  |-  ( E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  ( ( a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) )
34 df-3an 938 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) )  <->  ( ( a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) )
3533, 34bitr4i 244 . . . . . . . 8  |-  ( E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) ) )
3635exbii 1589 . . . . . . 7  |-  ( E. b E. x ( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  E. b
( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b
) ) )
3727, 36bitri 241 . . . . . 6  |-  ( E. x E. b ( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  E. b
( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b
) ) )
3837exbii 1589 . . . . 5  |-  ( E. a E. x E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  E. a E. b
( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b
) ) )
39 snex 4365 . . . . . 6  |-  { A }  e.  _V
40 uneq1 3454 . . . . . . 7  |-  ( a  =  A  ->  (
a  u.  b )  =  ( A  u.  b ) )
4140eqeq2d 2415 . . . . . 6  |-  ( a  =  A  ->  ( B  =  ( a  u.  b )  <->  B  =  ( A  u.  b
) ) )
42 uneq2 3455 . . . . . . 7  |-  ( b  =  { A }  ->  ( A  u.  b
)  =  ( A  u.  { A }
) )
4342eqeq2d 2415 . . . . . 6  |-  ( b  =  { A }  ->  ( B  =  ( A  u.  b )  <-> 
B  =  ( A  u.  { A }
) ) )
443, 39, 41, 43ceqsex2v 2953 . . . . 5  |-  ( E. a E. b ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) )  <->  B  =  ( A  u.  { A } ) )
4526, 38, 443bitri 263 . . . 4  |-  ( E. x E. a E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  B  =  ( A  u.  { A } ) )
465, 25, 453bitri 263 . . 3  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  B  =  ( A  u.  { A } ) )
47 df-suc 4547 . . . 4  |-  suc  A  =  ( A  u.  { A } )
4847eqeq2i 2414 . . 3  |-  ( B  =  suc  A  <->  B  =  ( A  u.  { A } ) )
4946, 48bitr4i 244 . 2  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  B  =  suc  A )
502, 49bitri 241 1  |-  ( ASucc B  <->  B  =  suc  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   {csn 3774   <.cop 3777   class class class wbr 4172    _I cid 4453   suc csuc 4543    o. ccom 4841    (x) ctxp 25587  Singletoncsingle 25595  Cupccup 25603  Succcsuccf 25605
This theorem is referenced by:  dfrdg4  25703
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-pow 4337  ax-pr 4363  ax-un 4660
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-mpt 4228  df-eprel 4454  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-1st 6308  df-2nd 6309  df-symdif 25576  df-txp 25609  df-singleton 25617  df-cup 25624  df-succf 25627
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