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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 4174 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | |
2 | ssun4 4175 | . 2 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | |
3 | 1, 2 | jaoi 855 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 ∪ cun 3946 ⊆ wss 3948 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 |
This theorem is referenced by: pwssun 5571 ordssun 6466 padct 31939 |
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