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| Mirrors > Home > MPE Home > Th. List > unieq | Structured version Visualization version GIF version | ||
| Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by BJ, 13-Apr-2024.) |
| Ref | Expression |
|---|---|
| unieq | ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqimss 4042 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
| 2 | 1 | unissd 4917 | . 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) |
| 3 | eqimss2 4043 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
| 4 | 3 | unissd 4917 | . 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐵 ⊆ ∪ 𝐴) |
| 5 | 2, 4 | eqssd 4001 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
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