Theorem 0npr 10416

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 2823	. 2 ⊢ ∅ = ∅
2		prn0 10413	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 3048	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  ∅c0 4293  Pcnp 10283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-np 10405

This theorem is referenced by:  genpass  10433  distrpr  10452  ltaddpr2  10459  ltapr  10469  addcanpr  10470  ltsrpr  10501  ltsosr  10518  mappsrpr  10532