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Theorem bj-intss 34285
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 5013 . . 3 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
21biimpi 217 . 2 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
3 intssuni 4889 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
4 sstr 3972 . . 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 3013  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-uni 4831  df-int 4868
This theorem is referenced by:  bj-0int  34287
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