Theorem intss2 5037

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 5029	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋)
2	1	biimpi 215	. 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋)
3		intssuni 4901	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4		sstr 3929	. . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋)
5	4	expcom 414	. 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 2943   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ 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-11 2154  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-uni 4840  df-int 4880

This theorem is referenced by:  intlidl  31602  bj-0int  35272