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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 4350 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssnpss 4080 | . 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ⊊ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊆ wss 3936 ⊊ wpss 3937 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ne 3017 df-dif 3939 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 |
This theorem is referenced by: pssnn 8730 pssn0 39106 |
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