Theorem 0npr 10906

Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
0npr	⊢ ¬ ∅ ∈ P

Proof of Theorem 0npr

Step	Hyp	Ref	Expression
1		eqid 2739	. 2 ⊢ ∅ = ∅
2		prn0 10903	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 2964	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119  ∅c0 4261  Pcnp 10773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-v 3433  df-dif 3886  df-ss 3900  df-pss 3903  df-nul 4262  df-np 10895

This theorem is referenced by:  genpass  10923  distrpr  10942  ltaddpr2  10949  ltapr  10959  addcanpr  10960  ltsrpr  10991  ltsosr  11008  mappsrpr  11022