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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 3924 | . 2 ⊢ ∅ ⊆ 𝐴 | |
2 | ssnpss 3672 | . 2 ⊢ (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ⊊ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 |
This theorem is referenced by: pssnn 8063 |
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