Theorem ssficl 40687

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 3414	. . 3 ⊢ 𝑥 ∈ V
3	2	inex1 5192	. 2 ⊢ (𝑥 ∩ 𝑦) ∈ V
4		sseq1 3920	. 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐵))
5		sseq1 3920	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
6		sseq1 3920	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
7		ssinss1 4145	. . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∩ 𝑦) ⊆ 𝐵)
8	7	adantr 484	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∩ 𝑦) ⊆ 𝐵)
9	1, 3, 4, 5, 6, 8	cllem0 40684	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2112  {cab 2736  ∀wral 3071  Vcvv 3410   ∩ cin 3860   ⊆ wss 3861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-v 3412  df-in 3868  df-ss 3878

This theorem is referenced by: (None)