![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4719 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) | |
2 | uniss 4681 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) | |
3 | 1, 2 | sylan9ssr 3841 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ≠ wne 2999 ⊆ wss 3798 ∅c0 4144 ∪ cuni 4658 ∩ cint 4697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 df-nul 4145 df-uni 4659 df-int 4698 |
This theorem is referenced by: rintn0 4840 fival 8587 mremre 16617 submre 16618 lssintcl 19323 iundifdifd 29916 iundifdif 29917 bj-ismoored2 33579 bj-ismooredr2 33581 ismrcd1 38098 |
Copyright terms: Public domain | W3C validator |