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| Mirrors > Home > MPE Home > Th. List > difcom | Structured version Visualization version GIF version | ||
| Description: Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.) | 
| Ref | Expression | 
|---|---|
| difcom | ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uncom 4157 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
| 2 | 1 | sseq2i 4012 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) | 
| 3 | ssundif 4487 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) | |
| 4 | ssundif 4487 | . 2 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
| 5 | 2, 3, 4 | 3bitr3i 301 | 1 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∖ cdif 3947 ∪ cun 3948 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 | 
| This theorem is referenced by: pssdifcom1 4489 pssdifcom2 4490 isreg2 23386 restmetu 24584 conss1 44468 icccncfext 45907 | 
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