Theorem prn0 10900

Description: A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prn0	⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)

Proof of Theorem prn0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnpi 10899	. . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl2 1193	. . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
3	1, 2	sylbi 217	. 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴)
4		0pss 4399	. 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	sylib 218	1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊊ wpss 3902  ∅c0 4285   class class class wbr 5098  Qcnq 10763   <_Q cltq 10769  Pcnp 10770

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-v 3442  df-dif 3904  df-ss 3918  df-pss 3921  df-nul 4286  df-np 10892

This theorem is referenced by:  0npr  10903  npomex  10907  genpn0  10914  prlem934  10944  ltaddpr  10945  prlem936  10958  reclem2pr  10959  suplem1pr  10963