Theorem superficl 43600

Description: The class of all supersets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)

Hypothesis

Ref	Expression
superficl.a	⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧}

Assertion

Ref	Expression
superficl	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵

Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem superficl

Step	Hyp	Ref	Expression
1		superficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧}
2		vex 3440	. . 3 ⊢ 𝑥 ∈ V
3	2	inex1 5250	. 2 ⊢ (𝑥 ∩ 𝑦) ∈ V
4		sseq2 3956	. 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦)))
5		sseq2 3956	. 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥))
6		sseq2 3956	. 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦))
7		ssin 4184	. . 3 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦))
8	7	biimpi 216	. 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∩ 𝑦))
9	1, 3, 4, 5, 6, 8	cllem0 43599	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-in 3904  df-ss 3914

This theorem is referenced by: (None)