![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > slelttrd | Structured version Visualization version GIF version |
Description: Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
slttrd.1 | âĒ (ð â ðī â No ) |
slttrd.2 | âĒ (ð â ðĩ â No ) |
slttrd.3 | âĒ (ð â ðķ â No ) |
slelttrd.4 | âĒ (ð â ðī âĪs ðĩ) |
slelttrd.5 | âĒ (ð â ðĩ <s ðķ) |
Ref | Expression |
---|---|
slelttrd | âĒ (ð â ðī <s ðķ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slelttrd.4 | . 2 âĒ (ð â ðī âĪs ðĩ) | |
2 | slelttrd.5 | . 2 âĒ (ð â ðĩ <s ðķ) | |
3 | slttrd.1 | . . 3 âĒ (ð â ðī â No ) | |
4 | slttrd.2 | . . 3 âĒ (ð â ðĩ â No ) | |
5 | slttrd.3 | . . 3 âĒ (ð â ðķ â No ) | |
6 | slelttr 27121 | . . 3 âĒ ((ðī â No ⧠ðĩ â No ⧠ðķ â No ) â ((ðī âĪs ðĩ ⧠ðĩ <s ðķ) â ðī <s ðķ)) | |
7 | 3, 4, 5, 6 | syl3anc 1372 | . 2 âĒ (ð â ((ðī âĪs ðĩ ⧠ðĩ <s ðķ) â ðī <s ðķ)) |
8 | 1, 2, 7 | mp2and 698 | 1 âĒ (ð â ðī <s ðķ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 ⧠wa 397 â wcel 2107 class class class wbr 5106 No csur 27004 <s cslt 27005 âĪs csle 27108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-1o 8413 df-2o 8414 df-no 27007 df-slt 27008 df-sle 27109 |
This theorem is referenced by: slerec 27180 sltlpss 27258 cofsslt 27259 sleadd1 27320 |
Copyright terms: Public domain | W3C validator |