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Mirrors > Home > MPE Home > Th. List > 0npr | Structured version Visualization version GIF version |
Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0npr | ⊢ ¬ ∅ ∈ P |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . 2 ⊢ ∅ = ∅ | |
2 | prn0 9690 | . . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅) | |
3 | 2 | necon2bi 2812 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ P |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ∅c0 3874 Pcnp 9560 |
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-3an 1033 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-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-np 9682 |
This theorem is referenced by: genpass 9710 distrpr 9729 ltaddpr2 9736 ltapr 9746 addcanpr 9747 ltsrpr 9777 ltsosr 9794 mappsrpr 9808 |
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