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Mirrors > Home > MPE Home > Th. List > intssuni2 | Structured version Visualization version GIF version |
Description: Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.) |
Ref | Expression |
---|---|
intssuni2 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intssuni 4860 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
2 | uniss 4808 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) | |
3 | 1, 2 | sylan9ssr 3929 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-uni 4801 df-int 4839 |
This theorem is referenced by: rintn0 4994 fival 8860 mremre 16867 submre 16868 lssintcl 19729 iundifdifd 30325 iundifdif 30326 bj-ismoored2 34523 bj-ismooredr2 34525 ismrcd1 39639 |
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