![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > incom | Structured version Visualization version GIF version |
Description: Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
incom | ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 454 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
2 | elin 4025 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
3 | elin 4025 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 295 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴)) |
5 | 4 | eqriv 2822 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
Copyright terms: Public domain | W3C validator |