| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > difuncomp | Structured version Visualization version GIF version | ||
| Description: Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.) |
| Ref | Expression |
|---|---|
| difuncomp | ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseqin2 4223 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐴) = 𝐴) | |
| 2 | 1 | biimpi 216 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝐶 ∩ 𝐴) = 𝐴) |
| 3 | incom 4209 | . . . 4 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶) | |
| 4 | 2, 3 | eqtr3di 2792 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 = (𝐴 ∩ 𝐶)) |
| 5 | 4 | difeq1d 4125 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = ((𝐴 ∩ 𝐶) ∖ 𝐵)) |
| 6 | difundi 4290 | . . . 4 ⊢ (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵)) = ((𝐶 ∖ (𝐶 ∖ 𝐴)) ∩ (𝐶 ∖ 𝐵)) | |
| 7 | dfss4 4269 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐶 ↔ (𝐶 ∖ (𝐶 ∖ 𝐴)) = 𝐴) | |
| 8 | 7 | biimpi 216 | . . . . 5 ⊢ (𝐴 ⊆ 𝐶 → (𝐶 ∖ (𝐶 ∖ 𝐴)) = 𝐴) |
| 9 | 8 | ineq1d 4219 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → ((𝐶 ∖ (𝐶 ∖ 𝐴)) ∩ (𝐶 ∖ 𝐵)) = (𝐴 ∩ (𝐶 ∖ 𝐵))) |
| 10 | 6, 9 | eqtrid 2789 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵)) = (𝐴 ∩ (𝐶 ∖ 𝐵))) |
| 11 | indif2 4281 | . . 3 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐵)) = ((𝐴 ∩ 𝐶) ∖ 𝐵) | |
| 12 | 10, 11 | eqtrdi 2793 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵)) = ((𝐴 ∩ 𝐶) ∖ 𝐵)) |
| 13 | 5, 12 | eqtr4d 2780 | 1 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = (𝐶 ∖ ((𝐶 ∖ 𝐴) ∪ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 |
| This theorem is referenced by: ldgenpisyslem1 34164 |
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