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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 5100 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋) |
| 3 | intssuni 4970 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
| 4 | sstr 3992 | . . 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 2940 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-pw 4602 df-uni 4908 df-int 4947 |
| This theorem is referenced by: intlidl 33448 bj-0int 37102 |
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