Theorem prpssnq 9812

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 9810	. 2 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
2		simpl3 1066	. 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 207	1 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊊ wpss 3575  ∅c0 3915   class class class wbr 4653  Qcnq 9674   <_Q cltq 9680  Pcnp 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pss 3590  df-np 9803

This theorem is referenced by:  elprnq  9813  npomex  9818  genpnnp  9827  prlem934  9855  ltexprlem2  9859  reclem2pr  9870  suplem1pr  9874  wuncn  9991