Theorem prpssnq 10730

Description: A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prpssnq	⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)

Proof of Theorem prpssnq

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

Step	Hyp	Ref	Expression
1		elnpi 10728	. 2 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl3 1191	. 2 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → 𝐴 ⊊ Q)
3	1, 2	sylbi 216	1 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ∀wal 1539   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066  Vcvv 3430   ⊊ wpss 3892  ∅c0 4261   class class class wbr 5078  Qcnq 10592   <_Q cltq 10598  Pcnp 10599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-ral 3070  df-rex 3071  df-v 3432  df-in 3898  df-ss 3908  df-pss 3910  df-np 10721

This theorem is referenced by:  elprnq  10731  npomex  10736  genpnnp  10745  prlem934  10773  ltexprlem2  10777  reclem2pr  10788  suplem1pr  10792  wuncn  10910