Theorem prpssnq 11068

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 11066	. 2 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl3 1212	. 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 220	1 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊊ wpss 3900  ∅c0 4279   class class class wbr 5103  Qcnq 10930   <_Q cltq 10936  Pcnp 10937

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-v 3453  df-ss 3916  df-pss 3919  df-np 11059

This theorem is used by:  elprnq  11069  npomex  11074  genpnnp  11083  prlem934  11111  ltexprlem2  11115  reclem2pr  11126  suplem1pr  11130  wuncn  11248