Theorem prn0 10795

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 10794	. . 3 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl2 1192	. . 3 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → ∅ ⊊ 𝐴)
3	1, 2	sylbi 216	. 2 ⊢ (𝐴 ∈ P → ∅ ⊊ 𝐴)
4		0pss 4384	. 2 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	sylib 217	1 ⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087  ∀wal 1537   ∈ wcel 2104   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3437   ⊊ wpss 3893  ∅c0 4262   class class class wbr 5081  Qcnq 10658   <_Q cltq 10664  Pcnp 10665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3439  df-dif 3895  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-np 10787

This theorem is referenced by:  0npr  10798  npomex  10802  genpn0  10809  prlem934  10839  ltaddpr  10840  prlem936  10853  reclem2pr  10854  suplem1pr  10858