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| Mirrors > Home > MPE Home > Th. List > uncom | Structured version Visualization version GIF version | ||
| Description: Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
| Ref | Expression |
|---|---|
| uncom | ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | orcom 871 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
| 2 | elun 4153 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐴) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)) | |
| 3 | 1, 2 | bitr4i 278 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐵 ∪ 𝐴)) |
| 4 | 3 | uneqri 4156 | 1 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
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