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Theorem intss2 4993
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 4985 . . 3 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
21biimpi 219 . 2 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
3 intssuni 4858 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
4 sstr 3885 . . 3 (( 𝐴 𝐴 𝐴𝑋) → 𝐴𝑋)
54expcom 417 . 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 2934  wss 3843  c0 4211  𝒫 cpw 4488   cuni 4796   cint 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212  df-pw 4490  df-uni 4797  df-int 4837
This theorem is referenced by:  intlidl  31174  bj-0int  34893
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