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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 10696 | . 2 ⊢ < ⊆ (ℝ* × ℝ*) | |
2 | relxp 5567 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 5650 | . 2 ⊢ ( < ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel < )) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel < |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3935 × cxp 5547 Rel wrel 5554 ℝ*cxr 10668 < clt 10669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-pr 4563 df-opab 5121 df-xp 5555 df-rel 5556 df-xr 10673 df-ltxr 10674 |
This theorem is referenced by: dflt2 12535 gtiso 30430 |
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