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

Theorem op1stb 4750
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5345 to extract the second member, op1sta 5343 for an alternate version, and op1st 6347 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 3975 . . . . 5  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43inteqi 4046 . . . 4  |-  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } }
5 snex 4397 . . . . . 6  |-  { A }  e.  _V
6 prex 4398 . . . . . 6  |-  { A ,  B }  e.  _V
75, 6intpr 4075 . . . . 5  |-  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
)
8 snsspr1 3939 . . . . . 6  |-  { A }  C_  { A ,  B }
9 df-ss 3326 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
108, 9mpbi 200 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
117, 10eqtri 2455 . . . 4  |-  |^| { { A } ,  { A ,  B } }  =  { A }
124, 11eqtri 2455 . . 3  |-  |^| <. A ,  B >.  =  { A }
1312inteqi 4046 . 2  |-  |^| |^| <. A ,  B >.  =  |^| { A }
141intsn 4078 . 2  |-  |^| { A }  =  A
1513, 14eqtri 2455 1  |-  |^| |^| <. A ,  B >.  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {csn 3806   {cpr 3807   <.cop 3809   |^|cint 4042
This theorem is referenced by:  elreldm  5086  op2ndb  5345  elxp5  5350  1stval2  6356  fundmen  7172  xpsnen  7184  xpnnenOLD  12799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-int 4043
  Copyright terms: Public domain W3C validator