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 | ||
|---|---|---|---|
| Assertion | unieq | |- ( A = B -> U. A = U. B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqimss | |- ( A = B -> A C_ B ) |
|
| 2 | 1 | unissd | |- ( A = B -> U. A C_ U. B ) |
| 3 | eqimss2 | |- ( A = B -> B C_ A ) |
|
| 4 | 3 | unissd | |- ( A = B -> U. B C_ U. A ) |
| 5 | 2 4 | eqssd | |- ( A = B -> U. A = U. B ) |