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Theorem brsuccf 24480
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 24413 . . 3  |- Succ  =  (Cup 
o.  (  _I  (x) Singleton ) )
21breqi 4029 . 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 4852 . . . 4  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  E. x
( A (  _I 
(x) Singleton ) x  /\  xCup B ) )
63brtxp2 24421 . . . . . . . . 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 2791 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
98ideq 4836 . . . . . . . . . . . . . 14  |-  ( A  _I  a  <->  A  =  a )
10 eqcom 2285 . . . . . . . . . . . . . 14  |-  ( A  =  a  <->  a  =  A )
119, 10bitri 240 . . . . . . . . . . . . 13  |-  ( A  _I  a  <->  a  =  A )
12 vex 2791 . . . . . . . . . . . . . 14  |-  b  e. 
_V
133, 12brsingle 24456 . . . . . . . . . . . . 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 1570 . . . . . . . . 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 1843 . . . . . . 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 1570 . . . . . 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 1569 . . . 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 1786 . . . . 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 1786 . . . . . . 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 4237 . . . . . . . . . 10  |-  <. a ,  b >.  e.  _V
29 breq1 4026 . . . . . . . . . . . 12  |-  ( x  =  <. a ,  b
>.  ->  ( xCup B  <->  <.
a ,  b >.Cup B ) )
308, 12, 4brcup 24478 . . . . . . . . . . . 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 2823 . . . . . . . . 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 1569 . . . . . . 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 1569 . . . . 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 4216 . . . . . 6  |-  { A }  e.  _V
40 uneq1 3322 . . . . . . 7  |-  ( a  =  A  ->  (
a  u.  b )  =  ( A  u.  b ) )
4140eqeq2d 2294 . . . . . 6  |-  ( a  =  A  ->  ( B  =  ( a  u.  b )  <->  B  =  ( A  u.  b
) ) )
42 uneq2 3323 . . . . . . 7  |-  ( b  =  { A }  ->  ( A  u.  b
)  =  ( A  u.  { A }
) )
4342eqeq2d 2294 . . . . . 6  |-  ( b  =  { A }  ->  ( B  =  ( A  u.  b )  <-> 
B  =  ( A  u.  { A }
) ) )
443, 39, 41, 43ceqsex2v 2825 . . . . 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 4398 . . . 4  |-  suc  A  =  ( A  u.  { A } )
4847eqeq2i 2293 . . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   <.cop 3643   class class class wbr 4023    _I cid 4304   suc csuc 4394    o. ccom 4693    (x) ctxp 24373  Singletoncsingle 24381  Cupccup 24389  Succcsuccf 24391
This theorem is referenced by:  dfrdg4  24488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-singleton 24403  df-cup 24410  df-succf 24413
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