Theorem bj-intss 33053

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
bj-intss	⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋))

Proof of Theorem bj-intss

Step	Hyp	Ref	Expression
1		sspwuni 4611	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋)
2	1	biimpi 206	. 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋)
3		intssuni 4499	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4		sstr 3611	. . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋)
5	4	expcom 451	. 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 2794   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ 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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476

This theorem is referenced by:  bj-0int  33055