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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 4175 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | eqeq1i 2823 | 1 ⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∩ cin 3932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-tru 1531 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-rab 3144 df-in 3940 |
This theorem is referenced by: ineqcomi 35386 |
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