Theorem ssficl 42870

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

Hypothesis

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

Assertion

Ref	Expression
ssficl	⊢ ∀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 ssficl

Step	Hyp	Ref	Expression
1		ssficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}
2		vex 3470	. . 3 ⊢ 𝑥 ∈ V
3	2	inex1 5308	. 2 ⊢ (𝑥 ∩ 𝑦) ∈ V
4		sseq1 4000	. 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐵))
5		sseq1 4000	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
6		sseq1 4000	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
7		ssinss1 4230	. . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∩ 𝑦) ⊆ 𝐵)
8	7	adantr 480	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∩ 𝑦) ⊆ 𝐵)
9	1, 3, 4, 5, 6, 8	cllem0 42867	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2098  {cab 2701  ∀wral 3053  Vcvv 3466   ∩ cin 3940   ⊆ wss 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-in 3948  df-ss 3958

This theorem is referenced by: (None)