Theorem 0npr 11101

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 2763	. 2 ⊢ ∅ = ∅
2		prn0 11098	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 2989	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  ∅c0 4281  Pcnp 10968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-nul 4282  df-np 11090

This theorem is used by:  genpass  11118  distrpr  11137  ltaddpr2  11144  ltapr  11154  addcanpr  11155  ltsrpr  11186  ltsosr  11203  mappsrpr  11217