Theorem ssuniint 42628

Description: Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
ssuniint.x	⊢ Ⅎ𝑥𝜑
ssuniint.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ssuniint.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)

Assertion

Ref	Expression
ssuniint	⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ssuniint

Step	Hyp	Ref	Expression
1		ssuniint.x	. . 3 ⊢ Ⅎ𝑥𝜑
2		ssuniint.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ssuniint.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)
4	1, 2, 3	elintd 42624	. 2 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
5		elssuni 4871	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 → 𝐴 ⊆ ∪ ∩ 𝐵)
6	4, 5	syl 17	1 ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396  Ⅎwnf 1786   ∈ wcel 2106   ⊆ wss 3887  ∪ cuni 4839  ∩ cint 4879

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-int 4880

This theorem is referenced by: (None)