Theorem 0npr 10403

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 2798	. 2 ⊢ ∅ = ∅
2		prn0 10400	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 3017	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  ∅c0 4243  Pcnp 10270

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-np 10392

This theorem is referenced by:  genpass  10420  distrpr  10439  ltaddpr2  10446  ltapr  10456  addcanpr  10457  ltsrpr  10488  ltsosr  10505  mappsrpr  10519