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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 5104 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
2 | 1 | biimpi 216 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋) |
3 | intssuni 4974 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
4 | sstr 4003 | . . 3 ⊢ ((∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋) → ∩ 𝐴 ⊆ 𝑋) | |
5 | 4 | expcom 413 | . 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 2937 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 ∪ cuni 4911 ∩ cint 4950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-v 3479 df-dif 3965 df-ss 3979 df-nul 4339 df-pw 4606 df-uni 4912 df-int 4951 |
This theorem is referenced by: intlidl 33427 bj-0int 37083 |
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