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Theorem op1stb 4568
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5154 to extract the second member, op1sta 5152 for an alternate version, and op1st 6089 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 3796 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 3867 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
5 snex 4215 . . . . . 6  |-  { A }  e.  _V
6 prex 4216 . . . . . 6  |-  { A ,  B }  e.  _V
75, 6intpr 3896 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
8 snsspr1 3765 . . . . . 6  |-  { A }  C_  { A ,  B }
9 df-ss 3167 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
108, 9mpbi 201 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
117, 10eqtri 2304 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
124, 11eqtri 2304 . . 3  |-  |^| <. A ,  B >.  =  { A }
1312inteqi 3867 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
141intsn 3899 . 2  |-  |^| { A }  =  A
1513, 14eqtri 2304 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1628    e. wcel 1688   _Vcvv 2789    i^i cin 3152    C_ wss 3153   {csn 3641   {cpr 3642   <.cop 3644   |^|cint 3863
This theorem is referenced by:  elreldm  4902  op2ndb  5154  elxp5  5159  1stval2  6098  fundmen  6929  xpsnen  6941  xpnnenOLD  12482
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-int 3864
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