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Theorem brsuccf 30700
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 30630 . . 3  |- Succ  =  (Cup 
o.  (  _I  (x) Singleton ) )
21breqi 4426 . 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 5020 . 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 4423 . . . . 5  |-  ( x  =  <. A ,  { A } >.  ->  ( xCup B  <->  <. A ,  { A } >.Cup B ) )
86, 7ceqsexv 3118 . . . 4  |-  ( E. x ( x  = 
<. A ,  { A } >.  /\  xCup B
)  <->  <. A ,  { A } >.Cup B )
9 snex 4658 . . . . 5  |-  { A }  e.  _V
103, 9, 4brcup 30698 . . . 4  |-  ( <. A ,  { A } >.Cup B  <->  B  =  ( A  u.  { A } ) )
118, 10bitri 252 . . 3  |-  ( E. x ( x  = 
<. A ,  { A } >.  /\  xCup B
)  <->  B  =  ( A  u.  { A } ) )
123brtxp2 30640 . . . . . 6  |-  ( A (  _I  (x) Singleton ) x  <->  E. a E. b ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
) )
1312anbi1i 699 . . . . 5  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  ( E. a E. b ( x  = 
<. a ,  b >.  /\  A  _I  a  /\  ASingleton b )  /\  xCup B ) )
14 3anass 986 . . . . . . . . 9  |-  ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  <->  ( x  = 
<. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) ) )
1514anbi1i 699 . . . . . . . 8  |-  ( ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( ( x  =  <. a ,  b
>.  /\  ( A  _I  a  /\  ASingleton b ) )  /\  xCup B ) )
16 an32 805 . . . . . . . 8  |-  ( ( ( x  =  <. a ,  b >.  /\  ( A  _I  a  /\  ASingleton b ) )  /\  xCup B )  <->  ( (
x  =  <. a ,  b >.  /\  xCup B )  /\  ( A  _I  a  /\  ASingleton b ) ) )
17 vex 3084 . . . . . . . . . . . . 13  |-  a  e. 
_V
1817ideq 5002 . . . . . . . . . . . 12  |-  ( A  _I  a  <->  A  =  a )
19 eqcom 2431 . . . . . . . . . . . 12  |-  ( A  =  a  <->  a  =  A )
2018, 19bitri 252 . . . . . . . . . . 11  |-  ( A  _I  a  <->  a  =  A )
21 vex 3084 . . . . . . . . . . . 12  |-  b  e. 
_V
223, 21brsingle 30676 . . . . . . . . . . 11  |-  ( ASingleton
b  <->  b  =  { A } )
2320, 22anbi12i 701 . . . . . . . . . 10  |-  ( ( A  _I  a  /\  ASingleton b )  <->  ( a  =  A  /\  b  =  { A } ) )
2423anbi1i 699 . . . . . . . . 9  |-  ( ( ( A  _I  a  /\  ASingleton b )  /\  ( x  =  <. a ,  b >.  /\  xCup B ) )  <->  ( (
a  =  A  /\  b  =  { A } )  /\  (
x  =  <. a ,  b >.  /\  xCup B ) ) )
25 ancom 451 . . . . . . . . 9  |-  ( ( ( x  =  <. a ,  b >.  /\  xCup B )  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( ( A  _I  a  /\  ASingleton b )  /\  (
x  =  <. a ,  b >.  /\  xCup B ) ) )
26 df-3an 984 . . . . . . . . 9  |-  ( ( a  =  A  /\  b  =  { A }  /\  ( x  = 
<. a ,  b >.  /\  xCup B ) )  <-> 
( ( a  =  A  /\  b  =  { A } )  /\  ( x  = 
<. a ,  b >.  /\  xCup B ) ) )
2724, 25, 263bitr4i 280 . . . . . . . 8  |-  ( ( ( x  =  <. a ,  b >.  /\  xCup B )  /\  ( A  _I  a  /\  ASingleton b ) )  <->  ( a  =  A  /\  b  =  { A }  /\  ( x  =  <. a ,  b >.  /\  xCup B ) ) )
2815, 16, 273bitri 274 . . . . . . 7  |-  ( ( ( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( a  =  A  /\  b  =  { A }  /\  ( x  =  <. a ,  b >.  /\  xCup B ) ) )
29282exbii 1713 . . . . . 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 1820 . . . . . 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 4184 . . . . . . . . 9  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
3231eqeq2d 2436 . . . . . . . 8  |-  ( a  =  A  ->  (
x  =  <. a ,  b >.  <->  x  =  <. A ,  b >.
) )
3332anbi1d 709 . . . . . . 7  |-  ( a  =  A  ->  (
( x  =  <. a ,  b >.  /\  xCup B )  <->  ( x  =  <. A ,  b
>.  /\  xCup B ) ) )
34 opeq2 4185 . . . . . . . . 9  |-  ( b  =  { A }  -> 
<. A ,  b >.  =  <. A ,  { A } >. )
3534eqeq2d 2436 . . . . . . . 8  |-  ( b  =  { A }  ->  ( x  =  <. A ,  b >.  <->  x  =  <. A ,  { A } >. ) )
3635anbi1d 709 . . . . . . 7  |-  ( b  =  { A }  ->  ( ( x  = 
<. A ,  b >.  /\  xCup B )  <->  ( x  =  <. A ,  { A } >.  /\  xCup B ) ) )
373, 9, 33, 36ceqsex2v 3120 . . . . . 6  |-  ( E. a E. b ( a  =  A  /\  b  =  { A }  /\  ( x  = 
<. a ,  b >.  /\  xCup B ) )  <-> 
( x  =  <. A ,  { A } >.  /\  xCup B ) )
3829, 30, 373bitr3i 278 . . . . 5  |-  ( ( E. a E. b
( x  =  <. a ,  b >.  /\  A  _I  a  /\  ASingleton b
)  /\  xCup B
)  <->  ( x  = 
<. A ,  { A } >.  /\  xCup B
) )
3913, 38bitri 252 . . . 4  |-  ( ( A (  _I  (x) Singleton ) x  /\  xCup B
)  <->  ( x  = 
<. A ,  { A } >.  /\  xCup B
) )
4039exbii 1712 . . 3  |-  ( E. x ( A (  _I  (x) Singleton ) x  /\  xCup B )  <->  E. x
( x  =  <. A ,  { A } >.  /\  xCup B ) )
41 df-suc 5444 . . . 4  |-  suc  A  =  ( A  u.  { A } )
4241eqeq2i 2440 . . 3  |-  ( B  =  suc  A  <->  B  =  ( A  u.  { A } ) )
4311, 40, 423bitr4i 280 . 2  |-  ( E. x ( A (  _I  (x) Singleton ) x  /\  xCup B )  <->  B  =  suc  A )
442, 5, 433bitri 274 1  |-  ( ASucc B  <->  B  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1868   _Vcvv 3081    u. cun 3434   {csn 3996   <.cop 4002   class class class wbr 4420    _I cid 4759    o. ccom 4853   suc csuc 5440    (x) ctxp 30588  Singletoncsingle 30596  Cupccup 30604  Succcsuccf 30606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-symdif 3693  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-eprel 4760  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fo 5603  df-fv 5605  df-1st 6803  df-2nd 6804  df-txp 30612  df-singleton 30620  df-cup 30627  df-succf 30630
This theorem is referenced by:  dfrdg4  30710
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