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Theorem brsuccf 30758
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 30688 . . 3  |- Succ  =  (Cup 
o.  (  _I  (x) Singleton ) )
21breqi 4424 . 2  |-  ( ASucc B  <->  A (Cup  o.  (  _I  (x) Singleton ) ) B )
3 brsuccf.1 . . 3  |-  A  e. 
_V
4 brsuccf.2 . . 3  |-  B  e. 
_V
53, 4brco 5027 . 2  |-  ( A (Cup  o.  (  _I 
(x) Singleton ) ) B  <->  E. x
( A (  _I 
(x) Singleton ) x  /\  xCup B ) )
6 opex 4681 . . . . 5  |-  <. A ,  { A } >.  e.  _V
7 breq1 4421 . . . . 5  |-  ( x  =  <. A ,  { A } >.  ->  ( xCup B  <->  <. A ,  { A } >.Cup B ) )
86, 7ceqsexv 3096 . . . 4  |-  ( E. x ( x  = 
<. A ,  { A } >.  /\  xCup B
)  <->  <. A ,  { A } >.Cup B )
9 snex 4658 . . . . 5  |-  { A }  e.  _V
103, 9, 4brcup 30756 . . . 4  |-  ( <. A ,  { A } >.Cup B  <->  B  =  ( A  u.  { A } ) )
118, 10bitri 257 . . 3  |-  ( E. x ( x  = 
<. A ,  { A } >.  /\  xCup B
)  <->  B  =  ( A  u.  { A } ) )
123brtxp2 30698 . . . . . 6  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
) )
1312anbi1i 706 . . . . 5  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  ( E. a E. b ( x  = 
<. a ,  b >.  /\  A  _I  a  /\  ASingleton b )  /\  xCup B ) )
14 3anass 995 . . . . . . . . 9  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) ) )
1514anbi1i 706 . . . . . . . 8  |-  ( ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( ( x  =  <. a ,  b
>.  /\  ( A  _I  a  /\  ASingleton b ) )  /\  xCup B ) )
16 an32 812 . . . . . . . 8  |-  ( ( ( x  =  <. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) )  /\  xCup B )  <->  ( (
x  =  <. a ,  b >.  /\  xCup B )  /\  ( A  _I  a  /\  ASingleton b ) ) )
17 vex 3060 . . . . . . . . . . . . 13  |-  a  e. 
_V
1817ideq 5009 . . . . . . . . . . . 12  |-  ( A  _I  a  <->  A  =  a )
19 eqcom 2469 . . . . . . . . . . . 12  |-  ( A  =  a  <->  a  =  A )
2018, 19bitri 257 . . . . . . . . . . 11  |-  ( A  _I  a  <->  a  =  A )
21 vex 3060 . . . . . . . . . . . 12  |-  b  e. 
_V
223, 21brsingle 30734 . . . . . . . . . . 11  |-  ( ASingleton
b  <->  b  =  { A } )
2320, 22anbi12i 708 . . . . . . . . . 10  |-  ( ( A  _I  a  /\  ASingleton b )  <->  ( a  =  A  /\  b  =  { A } ) )
2423anbi1i 706 . . . . . . . . 9  |-  ( ( ( A  _I  a  /\  ASingleton b )  /\  ( x  =  <. a ,  b >.  /\  xCup B ) )  <->  ( (
a  =  A  /\  b  =  { A } )  /\  (
x  =  <. a ,  b >.  /\  xCup B ) ) )
25 ancom 456 . . . . . . . . 9  |-  ( ( ( x  =  <. a ,  b >.  /\  xCup B )  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( ( A  _I  a  /\  ASingleton b )  /\  (
x  =  <. a ,  b >.  /\  xCup B ) ) )
26 df-3an 993 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { A }  /\  ( x  = 
<. a ,  b >.  /\  xCup B ) )  <-> 
( ( a  =  A  /\  b  =  { A } )  /\  ( x  = 
<. a ,  b >.  /\  xCup B ) ) )
2724, 25, 263bitr4i 285 . . . . . . . 8  |-  ( ( ( x  =  <. a ,  b >.  /\  xCup B )  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( a  =  A  /\  b  =  { A }  /\  ( x  =  <. a ,  b >.  /\  xCup B ) ) )
2815, 16, 273bitri 279 . . . . . . 7  |-  ( ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( a  =  A  /\  b  =  { A }  /\  ( x  =  <. a ,  b >.  /\  xCup B ) ) )
29282exbii 1730 . . . . . 6  |-  ( E. a E. b ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  E. a E. b
( a  =  A  /\  b  =  { A }  /\  (
x  =  <. a ,  b >.  /\  xCup B ) ) )
30 19.41vv 1842 . . . . . 6  |-  ( E. a E. b ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( E. a E. b ( x  = 
<. a ,  b >.  /\  A  _I  a  /\  ASingleton b )  /\  xCup B ) )
31 opeq1 4180 . . . . . . . . 9  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
3231eqeq2d 2472 . . . . . . . 8  |-  ( a  =  A  ->  (
x  =  <. a ,  b >.  <->  x  =  <. A ,  b >.
) )
3332anbi1d 716 . . . . . . 7  |-  ( a  =  A  ->  (
( x  =  <. a ,  b >.  /\  xCup B )  <->  ( x  =  <. A ,  b
>.  /\  xCup B ) ) )
34 opeq2 4181 . . . . . . . . 9  |-  ( b  =  { A }  -> 
<. A ,  b >.  =  <. A ,  { A } >. )
3534eqeq2d 2472 . . . . . . . 8  |-  ( b  =  { A }  ->  ( x  =  <. A ,  b >.  <->  x  =  <. A ,  { A } >. ) )
3635anbi1d 716 . . . . . . 7  |-  ( b  =  { A }  ->  ( ( x  = 
<. A ,  b >.  /\  xCup B )  <->  ( x  =  <. A ,  { A } >.  /\  xCup B ) ) )
373, 9, 33, 36ceqsex2v 3099 . . . . . 6  |-  ( E. a E. b ( a  =  A  /\  b  =  { A }  /\  ( x  = 
<. a ,  b >.  /\  xCup B ) )  <-> 
( x  =  <. A ,  { A } >.  /\  xCup B ) )
3829, 30, 373bitr3i 283 . . . . 5  |-  ( ( E. a E. b
( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( x  = 
<. A ,  { A } >.  /\  xCup B
) )
3913, 38bitri 257 . . . 4  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  ( x  = 
<. A ,  { A } >.  /\  xCup B
) )
4039exbii 1729 . . 3  |-  ( E. x ( A (  _I  (x) Singleton ) x  /\  xCup B )  <->  E. x
( x  =  <. A ,  { A } >.  /\  xCup B ) )
41 df-suc 5452 . . . 4  |-  suc  A  =  ( A  u.  { A } )
4241eqeq2i 2474 . . 3  |-  ( B  =  suc  A  <->  B  =  ( A  u.  { A } ) )
4311, 40, 423bitr4i 285 . 2  |-  ( E. x ( A (  _I  (x) Singleton ) x  /\  xCup B )  <->  B  =  suc  A )
442, 5, 433bitri 279 1  |-  ( ASucc B  <->  B  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   _Vcvv 3057    u. cun 3414   {csn 3980   <.cop 3986   class class class wbr 4418    _I cid 4766    o. ccom 4860   suc csuc 5448    (x) ctxp 30646  Singletoncsingle 30654  Cupccup 30662  Succcsuccf 30664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-symdif 3675  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-eprel 4767  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fo 5611  df-fv 5613  df-1st 6825  df-2nd 6826  df-txp 30670  df-singleton 30678  df-cup 30685  df-succf 30688
This theorem is referenced by:  dfrdg4  30768
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