Theorem prn0 10903

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 10902	. . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl2 1194	. . 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 4388	. 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	sylib 218	1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊊ wpss 3891  ∅c0 4274   class class class wbr 5086  Qcnq 10766   <_Q cltq 10772  Pcnp 10773

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-v 3432  df-dif 3893  df-ss 3907  df-pss 3910  df-nul 4275  df-np 10895

This theorem is referenced by:  0npr  10906  npomex  10910  genpn0  10917  prlem934  10947  ltaddpr  10948  prlem936  10961  reclem2pr  10962  suplem1pr  10966