MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op1stb Unicode version

Theorem op1stb 4569
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5156 to extract the second member, op1sta 5154 for an alternate version, and op1st 6128 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 3795 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 3866 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
5 snex 4216 . . . . . 6  |-  { A }  e.  _V
6 prex 4217 . . . . . 6  |-  { A ,  B }  e.  _V
75, 6intpr 3895 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
8 snsspr1 3764 . . . . . 6  |-  { A }  C_  { A ,  B }
9 df-ss 3166 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
108, 9mpbi 199 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
117, 10eqtri 2303 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
124, 11eqtri 2303 . . 3  |-  |^| <. A ,  B >.  =  { A }
1312inteqi 3866 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
141intsn 3898 . 2  |-  |^| { A }  =  A
1513, 14eqtri 2303 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   {cpr 3641   <.cop 3643   |^|cint 3862
This theorem is referenced by:  elreldm  4903  op2ndb  5156  elxp5  5161  1stval2  6137  fundmen  6934  xpsnen  6946  xpnnenOLD  12488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-int 3863
  Copyright terms: Public domain W3C validator