Theorem 0npr 10417

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 2824	. 2 ⊢ ∅ = ∅
2		prn0 10414	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 3049	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  ∅c0 4294  Pcnp 10284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-v 3499  df-dif 3942  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-np 10406

This theorem is referenced by:  genpass  10434  distrpr  10453  ltaddpr2  10460  ltapr  10470  addcanpr  10471  ltsrpr  10502  ltsosr  10519  mappsrpr  10533