Theorem 0npr 10915

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 2736	. 2 ⊢ ∅ = ∅
2		prn0 10912	. . 3 ⊢ (∅ ∈ P → ∅ ≠ ∅)
3	2	necon2bi 2962	. 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ P)
4	1, 3	ax-mp 5	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  ∅c0 4273  Pcnp 10782

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-v 3431  df-dif 3892  df-ss 3906  df-pss 3909  df-nul 4274  df-np 10904

This theorem is referenced by:  genpass  10932  distrpr  10951  ltaddpr2  10958  ltapr  10968  addcanpr  10969  ltsrpr  11000  ltsosr  11017  mappsrpr  11031