Theorem ssuniint 45826

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 45822	. 2 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
5		elssuni 4903	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 → 𝐴 ⊆ ∪ ∩ 𝐵)
6	4, 5	syl 18	1 ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  Ⅎwnf 1812   ∈ wcel 2142   ⊆ wss 3904  ∪ cuni 4871  ∩ cint 4911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3456  df-ss 3921  df-uni 4872  df-int 4912

This theorem is used by: (None)