Theorem op1stbg 4479

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 3776	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	inteqd 3849	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}})
3		snexg 4184	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
4		prexg 4211	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
5		intprg 3877	. . . . . 6 ⊢ (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
6	3, 4, 5	syl2an2r 595	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}))
7		snsspr1 3740	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
8		df-ss 3142	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
9	7, 8	mpbi 145	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
10	6, 9	eqtrdi 2226	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴})
11	2, 10	eqtrd 2210	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ⟨𝐴, 𝐵⟩ = {𝐴})
12	11	inteqd 3849	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴})
13		intsng 3878	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)
14	13	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴} = 𝐴)
15	12, 14	eqtrd 2210	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2737   ∩ cin 3128   ⊆ wss 3129  {csn 3592  {cpr 3593  ⟨cop 3595  ∩ cint 3844

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-int 3845

This theorem is referenced by:  elxp5  5117  fundmen  6805