Theorem op1stb 1992

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 2638 to extract the second member and op1sta 2635 for an alternate version.)

Hypothesis

Ref	Expression
intsn.1	⊢ A ∈ V

Assertion

Ref	Expression
op1stb	⊢ ∩∩⟨A, B⟩ = A

Proof of Theorem op1stb

Step	Hyp	Ref	Expression
1		df-op 1815	. . . . 5 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
2	1	inteqi 1969	. . . 4 ⊢ ∩⟨A, B⟩ = ∩{{A}, {A, B}}
3		snex 1859	. . . . 5 ⊢ {A} ∈ V
4		prex 1892	. . . . 5 ⊢ {A, B} ∈ V
5	3, 4	intpr 1990	. . . 4 ⊢ ∩{{A}, {A, B}} = ({A} ∩ {A, B})
6		snsspr 1853	. . . . 5 ⊢ {A} ⊆ {A, B}
7		df-ss 1492	. . . . 5 ⊢ ({A} ⊆ {A, B} ↔ ({A} ∩ {A, B}) = {A})
8	6, 7	mpbi 164	. . . 4 ⊢ ({A} ∩ {A, B}) = {A}
9	2, 5, 8	3eqtr 1123	. . 3 ⊢ ∩⟨A, B⟩ = {A}
10	9	inteqi 1969	. 2 ⊢ ∩∩⟨A, B⟩ = ∩{A}
11		intsn.1	. . 3 ⊢ A ∈ V
12	11	intsn 1991	. 2 ⊢ ∩{A} = A
13	10, 12	eqtr 1119	1 ⊢ ∩∩⟨A, B⟩ = A

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487  {csn 1808  {cpr 1809  ⟨cop 1810  ∩cint 1965

This theorem is referenced by:  elreldm 2554  op2ndb 2638  elxp5 2641  fundmen 3333  xpsnen 3339  mapunen 3397  xpnnen 4927

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-int 1966

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