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Mirrors > Home > MPE Home > Th. List > sseqin2 | Structured version Visualization version GIF version |
Description: A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.) |
Ref | Expression |
---|---|
sseqin2 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3861 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | incom 4092 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
3 | 2 | eqeq1i 2744 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
4 | 1, 3 | bitri 278 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
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