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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 10780 | . 2 ⊢ < ⊆ (ℝ* × ℝ*) | |
2 | relxp 5543 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 5627 | . 2 ⊢ ( < ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel < )) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel < |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3843 × cxp 5523 Rel wrel 5530 ℝ*cxr 10752 < clt 10753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-pr 4519 df-opab 5093 df-xp 5531 df-rel 5532 df-xr 10757 df-ltxr 10758 |
This theorem is referenced by: dflt2 12624 gtiso 30608 |
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