Theorem prn0 10413

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 10412	. . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl2 1188	. . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
3	1, 2	sylbi 219	. 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴)
4		0pss 4398	. 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	sylib 220	1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083  ∀wal 1535   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊊ wpss 3939  ∅c0 4293   class class class wbr 5068  Qcnq 10276   <_Q cltq 10282  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:  0npr  10416  npomex  10420  genpn0  10427  prlem934  10457  ltaddpr  10458  prlem936  10471  reclem2pr  10472  suplem1pr  10476