|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > elprnq | Structured version Visualization version GIF version | ||
| Description: A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| elprnq | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | prpssnq 11031 | . . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | |
| 2 | 1 | pssssd 4099 | . 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q) | 
| 3 | 2 | sselda 3982 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 Qcnq 10893 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: prub 11035 genpv 11040 genpdm 11043 genpss 11045 genpnnp 11046 genpnmax 11048 addclprlem1 11057 addclprlem2 11058 mulclprlem 11060 distrlem4pr 11067 1idpr 11070 psslinpr 11072 prlem934 11074 ltaddpr 11075 ltexprlem2 11078 ltexprlem3 11079 ltexprlem6 11082 ltexprlem7 11083 prlem936 11088 reclem2pr 11089 reclem3pr 11090 reclem4pr 11091 | 
| Copyright terms: Public domain | W3C validator |