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| Mirrors > Home > MPE Home > Th. List > undif | Structured version Visualization version GIF version | ||
| Description: Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.) |
| Ref | Expression |
|---|---|
| undif | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssequn1 4186 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | |
| 2 | undif2 4477 | . . 3 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
| 3 | 2 | eqeq1i 2742 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) |
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
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