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Theorem intss2 5112
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 5104 . . 3 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
21biimpi 216 . 2 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
3 intssuni 4974 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
4 sstr 4003 . . 3 (( 𝐴 𝐴 𝐴𝑋) → 𝐴𝑋)
54expcom 413 . 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 2937  wss 3962  c0 4338  𝒫 cpw 4604   cuni 4911   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-ss 3979  df-nul 4339  df-pw 4606  df-uni 4912  df-int 4951
This theorem is referenced by:  intlidl  33427  bj-0int  37083
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