Theorem op1stb 5440

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6228 to extract the second member, op1sta 6226 for an alternate version, and op1st 8009 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 4832	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 4911	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5		snex 5397	. . . . . 6 ⊢ {𝐴} ∈ V
6		prex 5396	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	intpr 4942	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8		snsspr1 4775	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		dfss2 3917	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
10	8, 9	mpbi 233	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
11	7, 10	eqtri 2784	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
12	4, 11	eqtri 2784	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
13	12	inteqi 4911	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
14	1	intsn 4944	. 2 ⊢ ∩ {𝐴} = 𝐴
15	13, 14	eqtri 2784	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  {csn 4584  {cpr 4586  ⟨cop 4590  ∩ cint 4907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-int 4908

This theorem is used by:  elreldm  5917  op2ndb  6228  elxp5  7935  1stval2  8018  fundmen  9059  xpsnen  9080