Theorem superficl 44711

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 3458	. . 3 ⊢ 𝑥 ∈ V
3	2	inex1 5281	. 2 ⊢ (𝑥 ∩ 𝑦) ∈ V
4		sseq2 3939	. 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦)))
5		sseq2 3939	. 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥))
6		sseq2 3939	. 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦))
7		ssin 4138	. . 3 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦))
8	7	biimpi 219	. 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∩ 𝑦))
9	1, 3, 4, 5, 6, 8	cllem0 44710	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-in 4094

This theorem is used by: (None)