Theorem op1stb 4514

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (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 3808	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 3879	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5	1	snex 4219	. . . . . 6 ⊢ {𝐴} ∈ V
6		prexg 4245	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 426	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
8	5, 7	intpr 3907	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
9		snsspr1 3771	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
10		df-ss 3170	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
11	9, 10	mpbi 145	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
12	8, 11	eqtri 2217	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
13	4, 12	eqtri 2217	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
14	13	inteqi 3879	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
15	1	intsn 3910	. 2 ⊢ ∩ {𝐴} = 𝐴
16	14, 15	eqtri 2217	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  {csn 3623  {cpr 3624  ⟨cop 3626  ∩ cint 3875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-int 3876

This theorem is referenced by:  elreldm  4893  op2ndb  5154  1stval2  6222  fundmen  6874  xpsnen  6889