Theorem op1stb 3136

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 3583 to extract the second member, op1sta 3579 for an alternate version, and op1st 4146 for the preferred version.)

Hypothesis

Ref	Expression
op1stb.1	|- A e. V

Assertion

Ref	Expression
op1stb	|- |^||^|<.A, B>. = A

Proof of Theorem op1stb

Step	Hyp	Ref	Expression
1		df-op 2474	. . . . 5 |- <.A, B>. = {{A}, {A, B}}
2	1	inteqi 2604	. . . 4 |- |^|<.A, B>. = |^|{{A}, {A, B}}
3		snex 2826	. . . . 5 |- {A} e. V
4		prex 2857	. . . . 5 |- {A, B} e. V
5	3, 4	intpr 2630	. . . 4 |- |^|{{A}, {A, B}} = ({A} i^i {A, B})
6		snsspr1 2534	. . . . 5 |- {A} (_ {A, B}
7		df-ss 2105	. . . . 5 |- ({A} (_ {A, B} <-> ({A} i^i {A, B}) = {A})
8	6, 7	mpbi 187	. . . 4 |- ({A} i^i {A, B}) = {A}
9	2, 5, 8	3eqtri 1542	. . 3 |- |^|<.A, B>. = {A}
10	9	inteqi 2604	. 2 |- |^||^|<.A, B>. = |^|{A}
11		op1stb.1	. . 3 |- A e. V
12	11	intsn 2631	. 2 |- |^|{A} = A
13	10, 12	eqtri 1538	1 |- |^||^|<.A, B>. = A

Syntax hints:   = wceq 992   e. wcel 994  Vcvv 1857   i^i cin 2098   (_ wss 2099  {csn 2467  {cpr 2468  <.cop 2469  |^|cint 2600

This theorem is referenced by:  elreldm 3425  op2ndb 3583  elxp5 3586  1stval2 4150  fundmen 4569  xpsnen 4576  mapunen 4649  xpnnen 7711

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-int 2601

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