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Mirrors > Home > MPE Home > Th. List > ltrelnq | Structured version Visualization version GIF version |
Description: Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltrelnq | ⊢ <Q ⊆ (Q × Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltnq 9619 | . 2 ⊢ <Q = ( <pQ ∩ (Q × Q)) | |
2 | inss2 3796 | . 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q) | |
3 | 1, 2 | eqsstri 3598 | 1 ⊢ <Q ⊆ (Q × Q) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3539 ⊆ wss 3540 × cxp 5036 <pQ cltpq 9551 Qcnq 9553 <Q cltq 9559 |
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-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-ltnq 9619 |
This theorem is referenced by: lterpq 9671 ltanq 9672 ltmnq 9673 ltexnq 9676 ltbtwnnq 9679 ltrnq 9680 prcdnq 9694 prnmadd 9698 genpcd 9707 nqpr 9715 1idpr 9730 prlem934 9734 ltexprlem4 9740 prlem936 9748 reclem2pr 9749 reclem3pr 9750 reclem4pr 9751 |
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