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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 4336 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssnpss 4043 | . 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ⊊ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊆ wss 3892 ⊊ wpss 3893 ∅c0 4262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ne 2946 df-v 3433 df-dif 3895 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 |
This theorem is referenced by: pssnn 8925 pssnnOLD 9010 pssn0 40191 |
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