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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 2737 | . 2 ⊢ ∅ = ∅ | |
| 2 | prn0 11029 | . . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅) | |
| 3 | 2 | necon2bi 2971 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P) |
| 4 | 1, 3 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ P |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∅c0 4333 Pcnp 10899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-pss 3971 df-nul 4334 df-np 11021 |
| This theorem is referenced by: genpass 11049 distrpr 11068 ltaddpr2 11075 ltapr 11085 addcanpr 11086 ltsrpr 11117 ltsosr 11134 mappsrpr 11148 |
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