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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 9691 | . . 3 ⊢ (𝐴 ∈ P → 𝐴 ⊊ Q) | |
2 | 1 | pssssd 3666 | . 2 ⊢ (𝐴 ∈ P → 𝐴 ⊆ Q) |
3 | 2 | sselda 3568 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Qcnq 9553 Pcnp 9560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-pss 3556 df-np 9682 |
This theorem is referenced by: prub 9695 genpv 9700 genpdm 9703 genpss 9705 genpnnp 9706 genpnmax 9708 addclprlem1 9717 addclprlem2 9718 mulclprlem 9720 distrlem4pr 9727 1idpr 9730 psslinpr 9732 prlem934 9734 ltaddpr 9735 ltexprlem2 9738 ltexprlem3 9739 ltexprlem6 9742 ltexprlem7 9743 prlem936 9748 reclem2pr 9749 reclem3pr 9750 reclem4pr 9751 |
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