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Theorem op1stb 4209
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 3548 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 3619 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
5 snexgOLD 3935 . . . . . . 7  |-  ( A  e.  _V  ->  { A }  e.  _V )
61, 5ax-mp 7 . . . . . 6  |-  { A }  e.  _V
7 prexgOLD 3946 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
81, 2, 7mp2an 402 . . . . . 6  |-  { A ,  B }  e.  _V
96, 8intpr 3647 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
10 snsspr1 3512 . . . . . 6  |-  { A }  C_  { A ,  B }
11 df-ss 2931 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
1210, 11mpbi 133 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
139, 12eqtri 2060 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
144, 13eqtri 2060 . . 3  |-  |^| <. A ,  B >.  =  { A }
1514inteqi 3619 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
161intsn 3650 . 2  |-  |^| { A }  =  A
1715, 16eqtri 2060 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = 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:  elreldm  4560  op2ndb  4804  1stval2  5782  fundmen  6286  xpsnen  6295
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