Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004)
Ref | Expression | ||
---|---|---|---|
Assertion | unundir | ⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unidm | ⊢ ( 𝐶 ∪ 𝐶 ) = 𝐶 | |
2 | 1 | uneq2i | ⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) |
3 | un4 | ⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐶 ) ) | |
4 | 2 3 | eqtr3i | ⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐶 ) ) |