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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-intss | Structured version Visualization version GIF version |
Description: A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.) |
Ref | Expression |
---|---|
bj-intss | ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwuni 5013 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
2 | 1 | biimpi 217 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋) |
3 | intssuni 4889 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | sstr 3972 | . . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋) | |
5 | 4 | expcom 414 | . 2 ⊢ (∪ 𝐴 ⊆ 𝑋 → (∩ 𝐴 ⊆ ∪ 𝐴 → ∩ 𝐴 ⊆ 𝑋)) |
6 | 2, 3, 5 | syl2im 40 | 1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋)) |
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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