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Mirrors > Home > MPE Home > Th. List > ssun | Structured version Visualization version GIF version |
Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.) |
Ref | Expression |
---|---|
ssun | ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun3 4150 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | |
2 | ssun4 4151 | . 2 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | |
3 | 1, 2 | jaoi 853 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∪ cun 3934 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-in 3943 df-ss 3952 |
This theorem is referenced by: pwunssOLD 5455 pwssun 5456 ordssun 6290 padct 30455 |
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