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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 2622 | . 2 ⊢ ∅ = ∅ | |
2 | prn0 9811 | . . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅) | |
3 | 2 | necon2bi 2824 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ P |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 ∅c0 3915 Pcnp 9681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-np 9803 |
This theorem is referenced by: genpass 9831 distrpr 9850 ltaddpr2 9857 ltapr 9867 addcanpr 9868 ltsrpr 9898 ltsosr 9915 mappsrpr 9929 |
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