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Theorem bj-intss 34395
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 ≠ ∅ → 𝐴𝑋))

Proof of Theorem bj-intss
StepHypRef Expression
1 sspwuni 5025 . . 3 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
21biimpi 218 . 2 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
3 intssuni 4901 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
4 sstr 3978 . . 3 (( 𝐴 𝐴 𝐴𝑋) → 𝐴𝑋)
54expcom 416 . 2 ( 𝐴𝑋 → ( 𝐴 𝐴 𝐴𝑋))
62, 3, 5syl2im 40 1 (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → 𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wne 3019  wss 3939  c0 4294  𝒫 cpw 4542   cuni 4841   cint 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-v 3499  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4544  df-uni 4842  df-int 4880
This theorem is referenced by:  bj-0int  34397
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