Theorem op1stb 5380

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6119 to extract the second member, op1sta 6117 for an alternate version, and op1st 7812 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 4800	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 4880	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5		snex 5349	. . . . . 6 ⊢ {𝐴} ∈ V
6		prex 5350	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	intpr 4910	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8		snsspr1 4744	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		df-ss 3900	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
10	8, 9	mpbi 229	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
11	7, 10	eqtri 2766	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
12	4, 11	eqtri 2766	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
13	12	inteqi 4880	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
14	1	intsn 4914	. 2 ⊢ ∩ {𝐴} = 𝐴
15	13, 14	eqtri 2766	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  {csn 4558  {cpr 4560  ⟨cop 4564  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-int 4877

This theorem is referenced by:  elreldm  5833  op2ndb  6119  elxp5  7744  1stval2  7821  fundmen  8775  xpsnen  8796