|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > prpssnq | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| prpssnq | ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elnpi 11029 | . 2 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))) | |
| 2 | simpl3 1193 | . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1537 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ⊊ wpss 3951 ∅c0 4332 class class class wbr 5142 Qcnq 10893 <Q cltq 10899 Pcnp 10900 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-v 3481 df-ss 3967 df-pss 3970 df-np 11022 | 
| This theorem is referenced by: elprnq 11032 npomex 11037 genpnnp 11046 prlem934 11074 ltexprlem2 11078 reclem2pr 11089 suplem1pr 11093 wuncn 11211 | 
| Copyright terms: Public domain | W3C validator |