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

Theorem op1stb 4722
 Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5497 to extract the second member, op1sta 5495 for an alternate version, and op1st 6808 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
Hypotheses
Ref Expression
op1stb.1
op1stb.2
Assertion
Ref Expression
op1stb

Proof of Theorem op1stb
StepHypRef Expression
1 op1stb.1 . . . . . 6
2 op1stb.2 . . . . . 6
31, 2dfop 4216 . . . . 5
43inteqi 4290 . . . 4
5 snex 4693 . . . . . 6
6 prex 4694 . . . . . 6
75, 6intpr 4320 . . . . 5
8 snsspr1 4179 . . . . . 6
9 df-ss 3489 . . . . . 6
108, 9mpbi 208 . . . . 5
117, 10eqtri 2486 . . . 4
124, 11eqtri 2486 . . 3
1312inteqi 4290 . 2
141intsn 4323 . 2
1513, 14eqtri 2486 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  e.wcel 1818   cvv 3109  i^icin 3474  C_wss 3475  {csn 4029  {cpr 4031  <.cop 4035  |^|cint 4286 This theorem is referenced by:  elreldm  5232  op2ndb  5497  elxp5  6745  1stval2  6817  fundmen  7609  xpsnen  7621  xpnnenOLD  13943 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-int 4287
 Copyright terms: Public domain W3C validator