Theorem 0npr 11004

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 11001	. . 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 1570   ∈ wcel 2145  ∅c0 4282  Pcnp 10871

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3455  df-dif 3905  df-ss 3919  df-pss 3922  df-nul 4283  df-np 10993

This theorem is used by:  genpass  11021  distrpr  11040  ltaddpr2  11047  ltapr  11057  addcanpr  11058  ltsrpr  11089  ltsosr  11106  mappsrpr  11120