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Mirrors > Home > MPE Home > Th. List > invdif | Structured version Visualization version GIF version |
Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
invdif | ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin2 4061 | . 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵))) | |
2 | ddif 3940 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
3 | 2 | difeq2i 3923 | . 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtri 2821 | 1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 Vcvv 3385 ∖ cdif 3766 ∩ cin 3768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rab 3098 df-v 3387 df-dif 3772 df-in 3776 |
This theorem is referenced by: indif2 4071 difundi 4080 difundir 4081 difindi 4082 difindir 4083 difdif2 4085 difun1 4088 undif1 4237 difdifdir 4250 frnsuppeq 7544 dfsup2 8592 fsets 16217 setsdm 16218 |
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