Theorem op1stb 4509

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 3803	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	inteqi 3874	. . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}}
5	1	snex 4214	. . . . . 6 ⊢ {𝐴} ∈ V
6		prexg 4240	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 426	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
8	5, 7	intpr 3902	. . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})
9		snsspr1 3766	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
10		df-ss 3166	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
11	9, 10	mpbi 145	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
12	8, 11	eqtri 2214	. . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}
13	4, 12	eqtri 2214	. . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴}
14	13	inteqi 3874	. 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴}
15	1	intsn 3905	. 2 ⊢ ∩ {𝐴} = 𝐴
16	14, 15	eqtri 2214	1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153  {csn 3618  {cpr 3619  ⟨cop 3621  ∩ cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-int 3871

This theorem is referenced by:  elreldm  4888  op2ndb  5149  1stval2  6208  fundmen  6860  xpsnen  6875