Theorem op1stb 5424

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6191 to extract the second member, op1sta 6189 for an alternate version, and op1st 7950 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 4815	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 4893	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5		snex 5381	. . . . . 6 ⊢ {𝐴} ∈ V
6		prex 5380	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	intpr 4924	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8		snsspr1 4757	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		dfss2 3907	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
10	8, 9	mpbi 230	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
11	7, 10	eqtri 2759	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
12	4, 11	eqtri 2759	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
13	12	inteqi 4893	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
14	1	intsn 4926	. 2 ⊢ ∩ {𝐴} = 𝐴
15	13, 14	eqtri 2759	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  {csn 4567  {cpr 4569  ⟨cop 4573  ∩ cint 4889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-int 4890

This theorem is referenced by:  elreldm  5890  op2ndb  6191  elxp5  7874  1stval2  7959  fundmen  8978  xpsnen  8999