Theorem op1stbg 4176

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)

Assertion

Ref	Expression
op1stbg	⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ ∩ ⟨A, B⟩ = A)

Proof of Theorem op1stbg

Step	Hyp	Ref	Expression
1		dfopg 3538	. . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ⟨A, B⟩ = {{A}, {A, B}})
2	1	inteqd 3611	. . . 4 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ ⟨A, B⟩ = ∩ {{A}, {A, B}})
3		elex 2560	. . . . . . . 8 ⊢ (A ∈ 𝑉 → A ∈ V)
4		snexgOLD 3926	. . . . . . . 8 ⊢ (A ∈ V → {A} ∈ V)
5	3, 4	syl 14	. . . . . . 7 ⊢ (A ∈ 𝑉 → {A} ∈ V)
6	5	adantr 261	. . . . . 6 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {A} ∈ V)
7		elex 2560	. . . . . . 7 ⊢ (B ∈ 𝑊 → B ∈ V)
8		prexgOLD 3937	. . . . . . 7 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
9	3, 7, 8	syl2an 273	. . . . . 6 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {A, B} ∈ V)
10		intprg 3639	. . . . . 6 ⊢ (({A} ∈ V ∧ {A, B} ∈ V) → ∩ {{A}, {A, B}} = ({A} ∩ {A, B}))
11	6, 9, 10	syl2anc 391	. . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ {{A}, {A, B}} = ({A} ∩ {A, B}))
12		snsspr1 3503	. . . . . 6 ⊢ {A} ⊆ {A, B}
13		df-ss 2925	. . . . . 6 ⊢ ({A} ⊆ {A, B} ↔ ({A} ∩ {A, B}) = {A})
14	12, 13	mpbi 133	. . . . 5 ⊢ ({A} ∩ {A, B}) = {A}
15	11, 14	syl6eq 2085	. . . 4 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ {{A}, {A, B}} = {A})
16	2, 15	eqtrd 2069	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ ⟨A, B⟩ = {A})
17	16	inteqd 3611	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ ∩ ⟨A, B⟩ = ∩ {A})
18		intsng 3640	. . 3 ⊢ (A ∈ 𝑉 → ∩ {A} = A)
19	18	adantr 261	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ {A} = A)
20	17, 19	eqtrd 2069	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∩ ∩ ⟨A, B⟩ = A)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   ⊆ wss 2911  {csn 3367  {cpr 3368  ⟨cop 3370  ∩ cint 3606

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607

This theorem is referenced by:  elxp5  4752  fundmen  6222