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Mirrors > Home > MPE Home > Th. List > difabs | Structured version Visualization version GIF version |
Description: Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.) |
Ref | Expression |
---|---|
difabs | ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difun1 3887 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = ((𝐴 ∖ 𝐵) ∖ 𝐵) | |
2 | unidm 3756 | . . 3 ⊢ (𝐵 ∪ 𝐵) = 𝐵 | |
3 | 2 | difeq2i 3725 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐵)) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtr3i 2646 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∖ cdif 3571 ∪ cun 3572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 |
This theorem is referenced by: axcclem 9279 lpdifsn 20947 compne 38643 compneOLD 38644 |
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