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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 4220 | . 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵))) | |
2 | ddif 4096 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
3 | 2 | difeq2i 4079 | . 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵) |
4 | 1, 3 | eqtri 2764 | 1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3445 ∖ cdif 3907 ∩ cin 3909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-in 3917 |
This theorem is referenced by: indif2 4230 difundi 4239 difundir 4240 difindi 4241 difindir 4242 difdif2 4246 difun1 4249 undif1 4435 difdifdir 4449 fsuppeq 8105 fsuppeqg 8106 dfsup2 9379 fsets 17040 setsdm 17041 |
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