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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 4970 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
| 2 | uniss 4915 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) | |
| 3 | 1, 2 | sylan9ssr 3998 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 ∪ 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-uni 4908 df-int 4947 | 
| This theorem is referenced by: rintn0 5109 fival 9452 mremre 17647 submre 17648 lssintcl 20962 iundifdifd 32574 iundifdif 32575 bj-ismoored2 37109 bj-ismooredr2 37111 ismrcd1 42709 | 
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