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Theorem op1stb 4506
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5108 to extract the second member, op1sta 5106 for an alternate version, and op1st 6027 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 3736 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 3807 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
5 snex 4154 . . . . . 6  |-  { A }  e.  _V
6 prex 4155 . . . . . 6  |-  { A ,  B }  e.  _V
75, 6intpr 3836 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
8 snsspr1 3705 . . . . . 6  |-  { A }  C_  { A ,  B }
9 df-ss 3108 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
108, 9mpbi 201 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
117, 10eqtri 2276 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
124, 11eqtri 2276 . . 3  |-  |^| <. A ,  B >.  =  { A }
1312inteqi 3807 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
141intsn 3839 . 2  |-  |^| { A }  =  A
1513, 14eqtri 2276 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2740    i^i cin 3093    C_ wss 3094   {csn 3581   {cpr 3582   <.cop 3584   |^|cint 3803
This theorem is referenced by:  elreldm  4856  op2ndb  5108  elxp5  5113  1stval2  6036  fundmen  6867  xpsnen  6879  xpnnenOLD  12415
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-int 3804
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