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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 4236 | . 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵))) | |
2 | ddif 4112 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
3 | 2 | difeq2i 4095 | . 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtri 2844 | 1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-in 3942 |
This theorem is referenced by: indif2 4246 difundi 4255 difundir 4256 difindi 4257 difindir 4258 difdif2 4260 difun1 4263 undif1 4423 difdifdir 4436 frnsuppeq 7836 dfsup2 8902 fsets 16510 setsdm 16511 |
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