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Mirrors > Home > MPE Home > Th. List > npss0 | Structured version Visualization version GIF version |
Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
npss0 | ⊢ ¬ 𝐴 ⊊ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4168 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssnpss 3907 | . 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ⊊ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊆ wss 3769 ⊊ wpss 3770 ∅c0 4115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-v 3387 df-dif 3772 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 |
This theorem is referenced by: pssnn 8420 pssn0 38035 |
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