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

Theorem op1stb 4585
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5343 to extract the second member, op1sta 5341 for an alternate version, and op1st 6591 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 4084 . . . . 5
43inteqi 4158 . . . 4
5 snex 4556 . . . . . 6
6 prex 4557 . . . . . 6
75, 6intpr 4187 . . . . 5
8 snsspr1 4047 . . . . . 6
9 df-ss 3379 . . . . . 6
108, 9mpbi 201 . . . . 5
117, 10eqtri 2509 . . . 4
124, 11eqtri 2509 . . 3
1312inteqi 4158 . 2
141intsn 4190 . 2
1513, 14eqtri 2509 1
Colors of variables: wff set class
Syntax hints:  =wceq 1670  e.wcel 1732   cvv 3015  i^icin 3364  C_wss 3365  {csn 3909  {cpr 3910  <.cop 3912  |^|cint 4154
This theorem is referenced by:  elreldm  5086  op2ndb  5343  elxp5  6531  1stval2  6600  fundmen  7345  xpsnen  7357  xpnnenOLD  13339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1570  ax-4 1581  ax-5 1644  ax-6 1685  ax-7 1705  ax-9 1736  ax-10 1751  ax-11 1756  ax-12 1768  ax-13 1955  ax-ext 2470  ax-sep 4439  ax-nul 4447  ax-pr 4554
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1338  df-ex 1566  df-nf 1569  df-sb 1677  df-clab 2476  df-cleq 2482  df-clel 2485  df-nfc 2614  df-ne 2654  df-ral 2764  df-v 3017  df-dif 3368  df-un 3370  df-in 3372  df-ss 3379  df-nul 3674  df-if 3826  df-sn 3915  df-pr 3916  df-op 3918  df-int 4155
  Copyright terms: Public domain W3C validator