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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 2738 | . 2 ⊢ ∅ = ∅ | |
2 | prn0 10859 | . . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅) | |
3 | 2 | necon2bi 2973 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ P |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ∅c0 4281 Pcnp 10729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-v 3446 df-dif 3912 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-np 10851 |
This theorem is referenced by: genpass 10879 distrpr 10898 ltaddpr2 10905 ltapr 10915 addcanpr 10916 ltsrpr 10947 ltsosr 10964 mappsrpr 10978 |
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