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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 4112 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | sseq2i 3972 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
3 | ssundif 4444 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) | |
4 | ssundif 4444 | . 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 205 ∖ cdif 3906 ∪ cun 3907 ⊆ wss 3909 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 |
This theorem is referenced by: pssdifcom1 4446 pssdifcom2 4447 isreg2 22680 restmetu 23878 conss1 42629 icccncfext 44023 |
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