ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  op1stb Unicode version

Theorem op1stb 4290
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)
Hypotheses
Ref Expression
op1stb.1  |-  A  e. 
_V
op1stb.2  |-  B  e. 
_V
Assertion
Ref Expression
op1stb  |-  |^| |^| <. A ,  B >.  =  A

Proof of Theorem op1stb
StepHypRef Expression
1 op1stb.1 . . . . . 6  |-  A  e. 
_V
2 op1stb.2 . . . . . 6  |-  B  e. 
_V
31, 2dfop 3616 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 3687 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
51snex 4011 . . . . . 6  |-  { A }  e.  _V
6 prexg 4029 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
71, 2, 6mp2an 417 . . . . . 6  |-  { A ,  B }  e.  _V
85, 7intpr 3715 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
9 snsspr1 3580 . . . . . 6  |-  { A }  C_  { A ,  B }
10 df-ss 3010 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
119, 10mpbi 143 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
128, 11eqtri 2108 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
134, 12eqtri 2108 . . 3  |-  |^| <. A ,  B >.  =  { A }
1413inteqi 3687 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
151intsn 3718 . 2  |-  |^| { A }  =  A
1614, 15eqtri 2108 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619    i^i cin 2996    C_ wss 2997   {csn 3441   {cpr 3442   <.cop 3444   |^|cint 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-int 3684
This theorem is referenced by:  elreldm  4649  op2ndb  4901  1stval2  5908  fundmen  6503  xpsnen  6517
  Copyright terms: Public domain W3C validator