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Mirrors > Home > MPE Home > Th. List > ltrel | Structured version Visualization version GIF version |
Description: "Less than" is a relation. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
ltrel | ⊢ Rel < |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelxr 10691 | . 2 ⊢ < ⊆ (ℝ* × ℝ*) | |
2 | relxp 5537 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 5620 | . 2 ⊢ ( < ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel < )) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel < |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3881 × cxp 5517 Rel wrel 5524 ℝ*cxr 10663 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pr 4528 df-opab 5093 df-xp 5525 df-rel 5526 df-xr 10668 df-ltxr 10669 |
This theorem is referenced by: dflt2 12529 gtiso 30460 |
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