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Mirrors > Home > MPE Home > Th. List > Mathboxes > ineqcom | Structured version Visualization version GIF version |
Description: Two ways of saying that two classes are disjoint (when 𝐶 = ∅: ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)). (Contributed by Peter Mazsa, 22-Mar-2017.) |
Ref | Expression |
---|---|
ineqcom | ⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4180 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | eqeq1i 2828 | 1 ⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∩ cin 3937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-rab 3149 df-in 3945 |
This theorem is referenced by: ineqcomi 35507 |
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