ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  op1stbg Unicode version

Theorem op1stbg 4210
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
Assertion
Ref Expression
op1stbg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )

Proof of Theorem op1stbg
StepHypRef Expression
1 dfopg 3547 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
21inteqd 3620 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } } )
3 elex 2566 . . . . . . . 8  |-  ( A  e.  V  ->  A  e.  _V )
4 snexgOLD 3935 . . . . . . . 8  |-  ( A  e.  _V  ->  { A }  e.  _V )
53, 4syl 14 . . . . . . 7  |-  ( A  e.  V  ->  { A }  e.  _V )
65adantr 261 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A }  e.  _V )
7 elex 2566 . . . . . . 7  |-  ( B  e.  W  ->  B  e.  _V )
8 prexgOLD 3946 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
93, 7, 8syl2an 273 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
10 intprg 3648 . . . . . 6  |-  ( ( { A }  e.  _V  /\  { A ,  B }  e.  _V )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
116, 9, 10syl2anc 391 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
12 snsspr1 3512 . . . . . 6  |-  { A }  C_  { A ,  B }
13 df-ss 2931 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
1412, 13mpbi 133 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
1511, 14syl6eq 2088 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  { A } )
162, 15eqtrd 2072 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  { A }
)
1716inteqd 3620 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  |^| { A } )
18 intsng 3649 . . 3  |-  ( A  e.  V  ->  |^| { A }  =  A )
1918adantr 261 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A }  =  A )
2017, 19eqtrd 2072 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2557    i^i cin 2916    C_ wss 2917   {csn 3375   {cpr 3376   <.cop 3378   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-int 3616
This theorem is referenced by:  elxp5  4809  fundmen  6286
  Copyright terms: Public domain W3C validator