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| Mirrors > Home > MPE Home > Th. List > un0 | Structured version Visualization version GIF version | ||
| Description: The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 15-Jul-1993.) |
| Ref | Expression |
|---|---|
| un0 | ⊢ (𝐴 ∪ ∅) = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel 4338 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
| 2 | 1 | biorfri 940 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅)) |
| 3 | 2 | bicomi 224 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅) ↔ 𝑥 ∈ 𝐴) |
| 4 | 3 | uneqri 4156 | 1 ⊢ (𝐴 ∪ ∅) = 𝐴 |
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