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Theorem brsuccf 27972
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 27902 . . 3  |- Succ  =  (Cup 
o.  (  _I  (x) Singleton ) )
21breqi 4298 . 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 5010 . . . 4  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  E. x
( A (  _I 
(x) Singleton ) x  /\  xCup B ) )
63brtxp2 27912 . . . . . . . . 9  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
) )
7 3anass 969 . . . . . . . . . . 11  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) ) )
8 vex 2975 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
98ideq 4992 . . . . . . . . . . . . . 14  |-  ( A  _I  a  <->  A  =  a )
10 eqcom 2445 . . . . . . . . . . . . . 14  |-  ( A  =  a  <->  a  =  A )
119, 10bitri 249 . . . . . . . . . . . . 13  |-  ( A  _I  a  <->  a  =  A )
12 vex 2975 . . . . . . . . . . . . . 14  |-  b  e. 
_V
133, 12brsingle 27948 . . . . . . . . . . . . 13  |-  ( ASingleton
b  <->  b  =  { A } )
1411, 13anbi12i 697 . . . . . . . . . . . 12  |-  ( ( A  _I  a  /\  ASingleton b )  <->  ( a  =  A  /\  b  =  { A } ) )
1514anbi2i 694 . . . . . . . . . . 11  |-  ( ( x  =  <. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( x  =  <. a ,  b
>.  /\  ( a  =  A  /\  b  =  { A } ) ) )
167, 15bitri 249 . . . . . . . . . 10  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) ) )
17162exbii 1635 . . . . . . . . 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 249 . . . . . . . 8  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) ) )
1918anbi1i 695 . . . . . . 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 252 . . . . . 6  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  E. a E. b
( ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) )  /\  xCup B ) )
22 anass 649 . . . . . . 7  |-  ( ( ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) )  /\  xCup B )  <->  ( x  =  <. a ,  b
>.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
23222exbii 1635 . . . . . 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 249 . . . . 5  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  E. a E. b
( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
2524exbii 1634 . . . 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 1787 . . . . 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 1787 . . . . . . 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 4556 . . . . . . . . . 10  |-  <. a ,  b >.  e.  _V
29 breq1 4295 . . . . . . . . . . . 12  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  <.
a ,  b >.Cup B ) )
308, 12, 4brcup 27970 . . . . . . . . . . . 12  |-  ( <.
a ,  b >.Cup B 
<->  B  =  ( a  u.  b ) )
3129, 30syl6bb 261 . . . . . . . . . . 11  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  B  =  ( a  u.  b ) ) )
3231anbi2d 703 . . . . . . . . . 10  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( a  =  A  /\  b  =  { A } )  /\  xCup B )  <->  ( (
a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) ) )
3328, 32ceqsexv 3009 . . . . . . . . 9  |-  ( E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  ( ( a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) )
34 df-3an 967 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) )  <->  ( ( a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) )
3533, 34bitr4i 252 . . . . . . . 8  |-  ( E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) ) )
3635exbii 1634 . . . . . . 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 249 . . . . . 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 1634 . . . . 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 4533 . . . . . 6  |-  { A }  e.  _V
40 uneq1 3503 . . . . . . 7  |-  ( a  =  A  ->  (
a  u.  b )  =  ( A  u.  b ) )
4140eqeq2d 2454 . . . . . 6  |-  ( a  =  A  ->  ( B  =  ( a  u.  b )  <->  B  =  ( A  u.  b
) ) )
42 uneq2 3504 . . . . . . 7  |-  ( b  =  { A }  ->  ( A  u.  b
)  =  ( A  u.  { A }
) )
4342eqeq2d 2454 . . . . . 6  |-  ( b  =  { A }  ->  ( B  =  ( A  u.  b )  <-> 
B  =  ( A  u.  { A }
) ) )
443, 39, 41, 43ceqsex2v 3011 . . . . 5  |-  ( E. a E. b ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) )  <->  B  =  ( A  u.  { A } ) )
4526, 38, 443bitri 271 . . . 4  |-  ( E. x E. a E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  B  =  ( A  u.  { A } ) )
465, 25, 453bitri 271 . . 3  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  B  =  ( A  u.  { A } ) )
47 df-suc 4725 . . . 4  |-  suc  A  =  ( A  u.  { A } )
4847eqeq2i 2453 . . 3  |-  ( B  =  suc  A  <->  B  =  ( A  u.  { A } ) )
4946, 48bitr4i 252 . 2  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  B  =  suc  A )
502, 49bitri 249 1  |-  ( ASucc B  <->  B  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2972    u. cun 3326   {csn 3877   <.cop 3883   class class class wbr 4292    _I cid 4631   suc csuc 4721    o. ccom 4844    (x) ctxp 27860  Singletoncsingle 27868  Cupccup 27876  Succcsuccf 27878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-eprel 4632  df-id 4636  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-1st 6577  df-2nd 6578  df-symdif 27849  df-txp 27884  df-singleton 27892  df-cup 27899  df-succf 27902
This theorem is referenced by:  dfrdg4  27981
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