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 2762	. 2 ⊢ ∅ = ∅
2		prn0 10980	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 2987	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142  ∅c0 4285  Pcnp 10850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3456  df-dif 3907  df-ss 3921  df-pss 3924  df-nul 4286  df-np 10972

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