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Mirrors > Home > MPE Home > Th. List > velcomp | Structured version Visualization version GIF version |
Description: Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.) |
Ref | Expression |
---|---|
velcomp | ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3475 | . 2 ⊢ 𝑥 ∈ V | |
2 | eldif 3957 | . 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | mpbiran 707 | 1 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2098 Vcvv 3471 ∖ cdif 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3473 df-dif 3950 |
This theorem is referenced by: rnep 5931 compeq 43880 compab 43882 |
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