Theorem intss2 5056

Description: A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.)

Assertion

Ref	Expression
intss2	⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋))

Proof of Theorem intss2

Step	Hyp	Ref	Expression
1		sspwuni 5048	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋)
2	1	biimpi 216	. 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋)
3		intssuni 4920	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4		sstr 3943	. . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋)
5	4	expcom 413	. 2 ⊢ (∪ 𝐴 ⊆ 𝑋 → (∩ 𝐴 ⊆ ∪ 𝐴 → ∩ 𝐴 ⊆ 𝑋))
6	2, 3, 5	syl2im 40	1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋))

Syntax hints:   → wi 4   ≠ wne 2928   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  ∪ cuni 4859  ∩ cint 4897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-dif 3905  df-ss 3919  df-nul 4284  df-pw 4552  df-uni 4860  df-int 4898

This theorem is referenced by:  intlidl  33380  bj-0int  37134