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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 4401 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssnpss 4102 | . 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ⊊ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊆ wss 3947 ⊊ wpss 3948 ∅c0 4325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-dif 3950 df-ss 3964 df-pss 3967 df-nul 4326 |
This theorem is referenced by: pssnn 9206 pssnnOLD 9299 pssn0 41949 |
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