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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 4980 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋) |
3 | intssuni 4853 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | sstr 3896 | . . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋) | |
5 | 4 | expcom 418 | . 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 2949 ⊆ wss 3854 ∅c0 4221 𝒫 cpw 4487 ∪ cuni 4791 ∩ cint 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-11 2159 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ne 2950 df-ral 3073 df-rex 3074 df-v 3409 df-dif 3857 df-in 3861 df-ss 3871 df-nul 4222 df-pw 4489 df-uni 4792 df-int 4832 |
This theorem is referenced by: intlidl 31108 bj-0int 34781 |
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