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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 4271 | . 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵))) | |
| 2 | ddif 4141 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
| 3 | 2 | difeq2i 4123 | . 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵) |
| 4 | 1, 3 | eqtri 2765 | 1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 |
| This theorem is referenced by: indif2 4281 difundi 4290 difundir 4291 difindi 4292 difindir 4293 difdif2 4296 difun1 4299 undif1 4476 difdifdir 4492 fsuppeq 8200 fsuppeqg 8201 dfsup2 9484 fsets 17206 setsdm 17207 |
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