Theorem op1stbg 4527

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 3817	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	inteqd 3890	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}})
3		snexg 4229	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
4		prexg 4256	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
5		intprg 3918	. . . . . 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 3781	. . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
8		df-ss 3179	. . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴})
9	7, 8	mpbi 145	. . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}
10	6, 9	eqtrdi 2254	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴})
11	2, 10	eqtrd 2238	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ⟨𝐴, 𝐵⟩ = {𝐴})
12	11	inteqd 3890	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴})
13		intsng 3919	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)
14	13	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴} = 𝐴)
15	12, 14	eqtrd 2238	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176  Vcvv 2772   ∩ cin 3165   ⊆ wss 3166  {csn 3633  {cpr 3634  ⟨cop 3636  ∩ cint 3885

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-int 3886

This theorem is referenced by:  elxp5  5172  fundmen  6900