Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10704 | . 2 ⊢ < ⊆ (ℝ* × ℝ*) | |
2 | relxp 5575 | . 2 ⊢ Rel (ℝ* × ℝ*) | |
3 | relss 5658 | . 2 ⊢ ( < ⊆ (ℝ* × ℝ*) → (Rel (ℝ* × ℝ*) → Rel < )) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel < |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3938 × cxp 5555 Rel wrel 5562 ℝ*cxr 10676 < clt 10677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-pr 4572 df-opab 5131 df-xp 5563 df-rel 5564 df-xr 10681 df-ltxr 10682 |
This theorem is referenced by: dflt2 12544 gtiso 30438 |
Copyright terms: Public domain | W3C validator |