Theorem op1stb 2913

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 3451 to extract the second member, op1sta 3448 for an alternate version, and op1st 4085 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 2416	. . . . 5 |- <.A, B>. = {{A}, {A, B}}
2	1	inteqi 2537	. . . 4 |- |^|<.A, B>. = |^|{{A}, {A, B}}
3		snex 2750	. . . . 5 |- {A} e. V
4		prex 2781	. . . . 5 |- {A, B} e. V
5	3, 4	intpr 2563	. . . 4 |- |^|{{A}, {A, B}} = ({A} i^i {A, B})
6		snsspr 2470	. . . . 5 |- {A} (_ {A, B}
7		df-ss 2053	. . . . 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 1499	. . 3 |- |^|<.A, B>. = {A}
10	9	inteqi 2537	. 2 |- |^||^|<.A, B>. = |^|{A}
11		op1stb.1	. . 3 |- A e. V
12	11	intsn 2564	. 2 |- |^|{A} = A
13	10, 12	eqtr 1495	1 |- |^||^|<.A, B>. = A

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811   i^i cin 2046   (_ wss 2047  {csn 2409  {cpr 2410  <.cop 2411  |^|cint 2533

This theorem is referenced by:  elreldm 3338  op2ndb 3451  elxp5 3454  1stval2 4089  fundmen 4428  xpsnen 4435  mapunen 4502  xpnnen 7499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-int 2534

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