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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 10959 | . 2 ⊢ <Q = ( <pQ ∩ (Q × Q)) | |
| 2 | inss2 4237 | . 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q) | |
| 3 | 1, 2 | eqsstri 4029 | 1 ⊢ <Q ⊆ (Q × Q) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∩ cin 3949 ⊆ wss 3950 × cxp 5682 <pQ cltpq 10891 Qcnq 10893 <Q cltq 10899 | 
| 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-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-in 3957 df-ss 3967 df-ltnq 10959 | 
| This theorem is referenced by: lterpq 11011 ltanq 11012 ltmnq 11013 ltexnq 11016 ltbtwnnq 11019 ltrnq 11020 prcdnq 11034 prnmadd 11038 genpcd 11047 nqpr 11055 1idpr 11070 prlem934 11074 ltexprlem4 11080 prlem936 11088 reclem2pr 11089 reclem3pr 11090 reclem4pr 11091 | 
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