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Theorem brsuccf 25038
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 24971 . . 3  |- Succ  =  (Cup 
o.  (  _I  (x) Singleton ) )
21breqi 4110 . 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 4934 . . . 4  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  E. x
( A (  _I 
(x) Singleton ) x  /\  xCup B ) )
63brtxp2 24979 . . . . . . . . 9  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
) )
7 3anass 938 . . . . . . . . . . 11  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) ) )
8 vex 2867 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
98ideq 4918 . . . . . . . . . . . . . 14  |-  ( A  _I  a  <->  A  =  a )
10 eqcom 2360 . . . . . . . . . . . . . 14  |-  ( A  =  a  <->  a  =  A )
119, 10bitri 240 . . . . . . . . . . . . 13  |-  ( A  _I  a  <->  a  =  A )
12 vex 2867 . . . . . . . . . . . . . 14  |-  b  e. 
_V
133, 12brsingle 25014 . . . . . . . . . . . . 13  |-  ( ASingleton
b  <->  b  =  { A } )
1411, 13anbi12i 678 . . . . . . . . . . . 12  |-  ( ( A  _I  a  /\  ASingleton b )  <->  ( a  =  A  /\  b  =  { A } ) )
1514anbi2i 675 . . . . . . . . . . 11  |-  ( ( x  =  <. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( x  =  <. a ,  b
>.  /\  ( a  =  A  /\  b  =  { A } ) ) )
167, 15bitri 240 . . . . . . . . . 10  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) ) )
17162exbii 1583 . . . . . . . . 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 240 . . . . . . . 8  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) ) )
1918anbi1i 676 . . . . . . 7  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  ( E. a E. b ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) )  /\  xCup B ) )
20 19.41vv 1907 . . . . . . 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 243 . . . . . 6  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  E. a E. b
( ( x  = 
<. a ,  b >.  /\  ( a  =  A  /\  b  =  { A } ) )  /\  xCup B ) )
22 anass 630 . . . . . . 7  |-  ( ( ( x  =  <. a ,  b >.  /\  (
a  =  A  /\  b  =  { A } ) )  /\  xCup B )  <->  ( x  =  <. a ,  b
>.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
23222exbii 1583 . . . . . 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 240 . . . . 5  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  E. a E. b
( x  =  <. a ,  b >.  /\  (
( a  =  A  /\  b  =  { A } )  /\  xCup B ) ) )
2524exbii 1582 . . . 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 1741 . . . . 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 1741 . . . . . . 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 4319 . . . . . . . . . 10  |-  <. a ,  b >.  e.  _V
29 breq1 4107 . . . . . . . . . . . 12  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  <.
a ,  b >.Cup B ) )
308, 12, 4brcup 25036 . . . . . . . . . . . 12  |-  ( <.
a ,  b >.Cup B 
<->  B  =  ( a  u.  b ) )
3129, 30syl6bb 252 . . . . . . . . . . 11  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  B  =  ( a  u.  b ) ) )
3231anbi2d 684 . . . . . . . . . 10  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( a  =  A  /\  b  =  { A } )  /\  xCup B )  <->  ( (
a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) ) )
3328, 32ceqsexv 2899 . . . . . . . . 9  |-  ( E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  ( ( a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) )
34 df-3an 936 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) )  <->  ( ( a  =  A  /\  b  =  { A } )  /\  B  =  ( a  u.  b ) ) )
3533, 34bitr4i 243 . . . . . . . 8  |-  ( E. x ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) ) )
3635exbii 1582 . . . . . . 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 240 . . . . . 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 1582 . . . . 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 4297 . . . . . 6  |-  { A }  e.  _V
40 uneq1 3398 . . . . . . 7  |-  ( a  =  A  ->  (
a  u.  b )  =  ( A  u.  b ) )
4140eqeq2d 2369 . . . . . 6  |-  ( a  =  A  ->  ( B  =  ( a  u.  b )  <->  B  =  ( A  u.  b
) ) )
42 uneq2 3399 . . . . . . 7  |-  ( b  =  { A }  ->  ( A  u.  b
)  =  ( A  u.  { A }
) )
4342eqeq2d 2369 . . . . . 6  |-  ( b  =  { A }  ->  ( B  =  ( A  u.  b )  <-> 
B  =  ( A  u.  { A }
) ) )
443, 39, 41, 43ceqsex2v 2901 . . . . 5  |-  ( E. a E. b ( a  =  A  /\  b  =  { A }  /\  B  =  ( a  u.  b ) )  <->  B  =  ( A  u.  { A } ) )
4526, 38, 443bitri 262 . . . 4  |-  ( E. x E. a E. b ( x  = 
<. a ,  b >.  /\  ( ( a  =  A  /\  b  =  { A } )  /\  xCup B ) )  <->  B  =  ( A  u.  { A } ) )
465, 25, 453bitri 262 . . 3  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  B  =  ( A  u.  { A } ) )
47 df-suc 4480 . . . 4  |-  suc  A  =  ( A  u.  { A } )
4847eqeq2i 2368 . . 3  |-  ( B  =  suc  A  <->  B  =  ( A  u.  { A } ) )
4946, 48bitr4i 243 . 2  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  B  =  suc  A )
502, 49bitri 240 1  |-  ( ASucc B  <->  B  =  suc  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226   {csn 3716   <.cop 3719   class class class wbr 4104    _I cid 4386   suc csuc 4476    o. ccom 4775    (x) ctxp 24931  Singletoncsingle 24939  Cupccup 24947  Succcsuccf 24949
This theorem is referenced by:  dfrdg4  25047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-eprel 4387  df-id 4391  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-1st 6209  df-2nd 6210  df-symdif 24920  df-txp 24953  df-singleton 24961  df-cup 24968  df-succf 24971
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