Theorem 0npr 10921

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 2729	. 2 ⊢ ∅ = ∅
2		prn0 10918	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 2955	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  ∅c0 4292  Pcnp 10788

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3446  df-dif 3914  df-ss 3928  df-pss 3931  df-nul 4293  df-np 10910

This theorem is referenced by:  genpass  10938  distrpr  10957  ltaddpr2  10964  ltapr  10974  addcanpr  10975  ltsrpr  11006  ltsosr  11023  mappsrpr  11037