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Theorem op1stb 4691
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5286 to extract the second member, op1sta 5284 for an alternate version, and op1st 6287 for the preferred version.) (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 3918 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 3989 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
5 snex 4339 . . . . . 6  |-  { A }  e.  _V
6 prex 4340 . . . . . 6  |-  { A ,  B }  e.  _V
75, 6intpr 4018 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
8 snsspr1 3883 . . . . . 6  |-  { A }  C_  { A ,  B }
9 df-ss 3270 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
108, 9mpbi 200 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
117, 10eqtri 2400 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
124, 11eqtri 2400 . . 3  |-  |^| <. A ,  B >.  =  { A }
1312inteqi 3989 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
141intsn 4021 . 2  |-  |^| { A }  =  A
1513, 14eqtri 2400 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2892    i^i cin 3255    C_ wss 3256   {csn 3750   {cpr 3751   <.cop 3753   |^|cint 3985
This theorem is referenced by:  elreldm  5027  op2ndb  5286  elxp5  5291  1stval2  6296  fundmen  7109  xpsnen  7121  xpnnenOLD  12729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-int 3986
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