Theorem prpssnq 10892

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 10890	. 2 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl3 1194	. 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 217	1 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ⊊ wpss 3899  ∅c0 4282   class class class wbr 5095  Qcnq 10754   <_Q cltq 10760  Pcnp 10761

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-v 3439  df-ss 3915  df-pss 3918  df-np 10883

This theorem is referenced by:  elprnq  10893  npomex  10898  genpnnp  10907  prlem934  10935  ltexprlem2  10939  reclem2pr  10950  suplem1pr  10954  wuncn  11072