Theorem op1stb 5414

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6182 to extract the second member, op1sta 6180 for an alternate version, and op1st 7943 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

Hypotheses

Ref	Expression
op1stb.1	⊢ 𝐴 ∈ V
op1stb.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
op1stb	⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Proof of Theorem op1stb

Step	Hyp	Ref	Expression
1		op1stb.1	. . . . . 6 ⊢ 𝐴 ∈ V
2		op1stb.2	. . . . . 6 ⊢ 𝐵 ∈ V
3	1, 2	dfop 4806	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 4884	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5		snex 5371	. . . . . 6 ⊢ {𝐴} ∈ V
6		prex 5370	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	intpr 4915	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8		snsspr1 4748	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		dfss2 3903	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
10	8, 9	mpbi 232	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
11	7, 10	eqtri 2764	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
12	4, 11	eqtri 2764	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
13	12	inteqi 4884	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
14	1	intsn 4917	. 2 ⊢ ∩ {𝐴} = 𝐴
15	13, 14	eqtri 2764	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Syntax hints:   = wceq 1548   ∈ wcel 2121  Vcvv 3433   ∩ cin 3884   ⊆ wss 3885  {csn 4558  {cpr 4560  ⟨cop 4564  ∩ cint 4880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-int 4881

This theorem is referenced by:  elreldm  5884  op2ndb  6182  elxp5  7867  1stval2  7952  fundmen  8972  xpsnen  8993