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Mirrors > Home > MPE Home > Th. List > intss2 | 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 |
---|---|
intss2 | ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwuni 4985 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋) |
3 | intssuni 4858 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | sstr 3885 | . . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋) | |
5 | 4 | expcom 417 | . 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 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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