Theorem prn0 10998

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 10997	. . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl2 1211	. . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
3	1, 2	sylbi 220	. 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴)
4		0pss 4360	. 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	sylib 221	1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊊ wpss 3900  ∅c0 4279   class class class wbr 5103  Qcnq 10861   <_Q cltq 10867  Pcnp 10868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-v 3452  df-dif 3902  df-ss 3916  df-pss 3919  df-nul 4280  df-np 10990

This theorem is used by:  0npr  11001  npomex  11005  genpn0  11012  prlem934  11042  ltaddpr  11043  prlem936  11056  reclem2pr  11057  suplem1pr  11061