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| Mirrors > Home > MPE Home > Th. List > unssd | Structured version Visualization version GIF version | ||
| Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
| Ref | Expression |
|---|---|
| unssd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| unssd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
| Ref | Expression |
|---|---|
| unssd | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unssd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
| 2 | unssd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
| 3 | unss 4190 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
| 4 | 3 | biimpi 216 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
| 5 | 1, 2, 4 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
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