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Mirrors > Home > MPE Home > Th. List > Mathboxes > compel | Structured version Visualization version GIF version |
Description: Equivalence between two ways of saying "is a member of the complement of 𝐴." (Contributed by Andrew Salmon, 15-Jul-2011.) |
Ref | Expression |
---|---|
compel | ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3307 | . 2 ⊢ 𝑥 ∈ V | |
2 | eldif 3690 | . 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | mpbiran 991 | 1 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∈ wcel 2103 Vcvv 3304 ∖ cdif 3677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-v 3306 df-dif 3683 |
This theorem is referenced by: compeq 39061 compab 39064 conss34OLD 39065 |
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