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

Step	Hyp	Ref	Expression
1		op1stb.1	. . . . . 6
2		op1stb.2	. . . . . 6
3	1, 2	dfop 4216	. . . . 5
4	3	inteqi 4290	. . . 4
5		snex 4693	. . . . . 6
6		prex 4694	. . . . . 6
7	5, 6	intpr 4320	. . . . 5
8		snsspr1 4179	. . . . . 6
9		df-ss 3489	. . . . . 6
10	8, 9	mpbi 208	. . . . 5
11	7, 10	eqtri 2486	. . . 4
12	4, 11	eqtri 2486	. . 3
13	12	inteqi 4290	. 2
14	1	intsn 4323	. 2
15	13, 14	eqtri 2486	1

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