Theorem op1stb 5439

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6217 to extract the second member, op1sta 6215 for an alternate version, and op1st 7992 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 4831	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 4910	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5		snex 5396	. . . . . 6 ⊢ {𝐴} ∈ V
6		prex 5395	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	intpr 4941	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
8		snsspr1 4774	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		dfss2 3916	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
10	8, 9	mpbi 233	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
11	7, 10	eqtri 2783	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
12	4, 11	eqtri 2783	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
13	12	inteqi 4910	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
14	1	intsn 4943	. 2 ⊢ ∩ {𝐴} = 𝐴
15	13, 14	eqtri 2783	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898  {csn 4583  {cpr 4585  ⟨cop 4589  ∩ cint 4906

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-int 4907

This theorem is used by:  elreldm  5913  op2ndb  6217  elxp5  7918  1stval2  8001  fundmen  9037  xpsnen  9058