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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 ≠ ∅ → 𝐴𝑋))

Proof of Theorem intss2
StepHypRef Expression
1 sspwuni 5029 . . 3 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
21biimpi 215 . 2 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
3 intssuni 4901 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
4 sstr 3929 . . 3 (( 𝐴 𝐴 𝐴𝑋) → 𝐴𝑋)
54expcom 414 . 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 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
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