Theorem op1stb 2903

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 3437 to extract the second member, op1sta 3434 for an alternate version, and op1st 4069 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 2406	. . . . 5 |- <.A, B>. = {{A}, {A, B}}
2	1	inteqi 2527	. . . 4 |- |^|<.A, B>. = |^|{{A}, {A, B}}
3		snex 2740	. . . . 5 |- {A} e. V
4		prex 2771	. . . . 5 |- {A, B} e. V
5	3, 4	intpr 2553	. . . 4 |- |^|{{A}, {A, B}} = ({A} i^i {A, B})
6		snsspr 2461	. . . . 5 |- {A} (_ {A, B}
7		df-ss 2043	. . . . 5 |- ({A} (_ {A, B} <-> ({A} i^i {A, B}) = {A})
8	6, 7	mpbi 189	. . . 4 |- ({A} i^i {A, B}) = {A}
9	2, 5, 8	3eqtr 1491	. . 3 |- |^|<.A, B>. = {A}
10	9	inteqi 2527	. 2 |- |^||^|<.A, B>. = |^|{A}
11		op1stb.1	. . 3 |- A e. V
12	11	intsn 2554	. 2 |- |^|{A} = A
13	10, 12	eqtr 1487	1 |- |^||^|<.A, B>. = A

Syntax hints:   = wceq 953   e. wcel 955  Vcvv 1802   i^i cin 2036   (_ wss 2037  {csn 2399  {cpr 2400  <.cop 2401  |^|cint 2523

This theorem is referenced by:  elreldm 3327  op2ndb 3437  elxp5 3440  1stval2 4073  fundmen 4409  xpsnen 4415  mapunen 4482  xpnnen 7441

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-int 2524

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