Theorem ssuniint 39250

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 39245	. 2 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
5		elssuni 4467	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 → 𝐴 ⊆ ∪ ∩ 𝐵)
6	4, 5	syl 17	1 ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵)

Syntax hints:   → wi 4   ∧ wa 384  Ⅎwnf 1708   ∈ wcel 1990   ⊆ wss 3574  ∪ cuni 4436  ∩ cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-int 4476

This theorem is referenced by: (None)