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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 27710 | . . 3 âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī âĪs ðĩ â§ ðĩ <s ðķ) â ðī <s ðķ)) | |
7 | 3, 4, 5, 6 | syl3anc 1368 | . 2 âĒ (ð â ((ðī âĪs ðĩ â§ ðĩ <s 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 â§ wa 394 â wcel 2098 class class class wbr 5152 No csur 27593 <s cslt 27594 âĪs csle 27697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-1o 8493 df-2o 8494 df-no 27596 df-slt 27597 df-sle 27698 |
This theorem is referenced by: slerec 27772 sltlpss 27853 cofsslt 27858 sleadd1 27926 sltmul12ad 28103 absslt 28163 |
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