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Mirrors > Home > MPE Home > Th. List > sltletrd | 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 ) |
sltletrd.4 | âĒ (ð â ðī <s ðĩ) |
sltletrd.5 | âĒ (ð â ðĩ âĪs ðķ) |
Ref | Expression |
---|---|
sltletrd | âĒ (ð â ðī <s ðķ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltletrd.4 | . 2 âĒ (ð â ðī <s ðĩ) | |
2 | sltletrd.5 | . 2 âĒ (ð â ðĩ âĪs 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 | sltletr 27056 | . . 3 âĒ ((ðī â No ⧠ðĩ â No ⧠ðķ â No ) â ((ðī <s ðĩ ⧠ðĩ âĪs ðķ) â ðī <s ðķ)) | |
7 | 3, 4, 5, 6 | syl3anc 1371 | . 2 âĒ (ð â ((ðī <s ðĩ ⧠ðĩ âĪs ðķ) â ðī <s ðķ)) |
8 | 1, 2, 7 | mp2and 697 | 1 âĒ (ð â ðī <s ðķ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 ⧠wa 396 â wcel 2106 class class class wbr 5103 No csur 26940 <s cslt 26941 âĪs csle 27044 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-1o 8404 df-2o 8405 df-no 26943 df-slt 26944 df-sle 27045 |
This theorem is referenced by: slerec 27110 coinitsslt 27187 sleadd1 34301 |
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