Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4053 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | sseq2i 3916 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
3 | ssundif 4385 | . 2 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) | |
4 | ssundif 4385 | . 2 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
5 | 2, 3, 4 | 3bitr3i 304 | 1 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∖ cdif 3850 ∪ cun 3851 ⊆ wss 3853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 |
This theorem is referenced by: pssdifcom1 4387 pssdifcom2 4388 isreg2 22228 restmetu 23422 conss1 41676 icccncfext 43046 |
Copyright terms: Public domain | W3C validator |