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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > indifbi | Structured version Visualization version GIF version |
Description: Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
Ref | Expression |
---|---|
indifbi | ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4129 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | inss1 4129 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴 | |
3 | rcompleq 4196 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∩ 𝐶)))) | |
4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∩ 𝐶))) |
5 | difin 4162 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
6 | difin 4162 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐶)) = (𝐴 ∖ 𝐶) | |
7 | 5, 6 | eqeq12i 2774 | . 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∩ 𝐶)) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) |
8 | 4, 7 | bitri 278 | 1 ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1539 ∖ cdif 3851 ∩ cin 3853 ⊆ wss 3854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-dif 3857 df-in 3861 df-ss 3871 |
This theorem is referenced by: fressupp 30531 |
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