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Theorem bj-intss 33280
 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 4719 . . 3 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
21biimpi 206 . 2 (𝐴 ⊆ 𝒫 𝑋 𝐴𝑋)
3 intssuni 4607 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
4 sstr 3717 . . 3 (( 𝐴 𝐴 𝐴𝑋) → 𝐴𝑋)
54expcom 450 . 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 2896   ⊆ wss 3680  ∅c0 4023  𝒫 cpw 4266  ∪ cuni 4544  ∩ cint 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-nul 4024  df-pw 4268  df-uni 4545  df-int 4584 This theorem is referenced by:  bj-0int  33282
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