Theorem 0npr 10983

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 2732	. 2 ⊢ ∅ = ∅
2		prn0 10980	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 2971	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  ∅c0 4321  Pcnp 10850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-np 10972

This theorem is referenced by:  genpass  11000  distrpr  11019  ltaddpr2  11026  ltapr  11036  addcanpr  11037  ltsrpr  11068  ltsosr  11085  mappsrpr  11099