![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sltlend | Structured version Visualization version GIF version |
Description: Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 15-Apr-2025.) |
Ref | Expression |
---|---|
sltlen.1 | âĒ (ð â ðī â No ) |
sltlen.2 | âĒ (ð â ðĩ â No ) |
Ref | Expression |
---|---|
sltlend | âĒ (ð â (ðī <s ðĩ â (ðī âĪs ðĩ â§ ðĩ â ðī))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltlen.1 | . . . . . 6 âĒ (ð â ðī â No ) | |
2 | 1 | adantr 480 | . . . . 5 âĒ ((ð â§ ðī <s ðĩ) â ðī â No ) |
3 | sltlen.2 | . . . . . 6 âĒ (ð â ðĩ â No ) | |
4 | 3 | adantr 480 | . . . . 5 âĒ ((ð â§ ðī <s ðĩ) â ðĩ â No ) |
5 | simpr 484 | . . . . 5 âĒ ((ð â§ ðī <s ðĩ) â ðī <s ðĩ) | |
6 | 2, 4, 5 | sltled 27696 | . . . 4 âĒ ((ð â§ ðī <s ðĩ) â ðī âĪs ðĩ) |
7 | 6 | ex 412 | . . 3 âĒ (ð â (ðī <s ðĩ â ðī âĪs ðĩ)) |
8 | sltne 27697 | . . . . 5 âĒ ((ðī â No â§ ðī <s ðĩ) â ðĩ â ðī) | |
9 | 1, 8 | sylan 579 | . . . 4 âĒ ((ð â§ ðī <s ðĩ) â ðĩ â ðī) |
10 | 9 | ex 412 | . . 3 âĒ (ð â (ðī <s ðĩ â ðĩ â ðī)) |
11 | 7, 10 | jcad 512 | . 2 âĒ (ð â (ðī <s ðĩ â (ðī âĪs ðĩ â§ ðĩ â ðī))) |
12 | sleloe 27681 | . . . . 5 âĒ ((ðī â No â§ ðĩ â No ) â (ðī âĪs ðĩ â (ðī <s ðĩ âĻ ðī = ðĩ))) | |
13 | 1, 3, 12 | syl2anc 583 | . . . 4 âĒ (ð â (ðī âĪs ðĩ â (ðī <s ðĩ âĻ ðī = ðĩ))) |
14 | eqneqall 2947 | . . . . . 6 âĒ (ðĩ = ðī â (ðĩ â ðī â ðī <s ðĩ)) | |
15 | 14 | eqcoms 2736 | . . . . 5 âĒ (ðī = ðĩ â (ðĩ â ðī â ðī <s ðĩ)) |
16 | 15 | jao1i 857 | . . . 4 âĒ ((ðī <s ðĩ âĻ ðī = ðĩ) â (ðĩ â ðī â ðī <s ðĩ)) |
17 | 13, 16 | biimtrdi 252 | . . 3 âĒ (ð â (ðī âĪs ðĩ â (ðĩ â ðī â ðī <s ðĩ))) |
18 | 17 | impd 410 | . 2 âĒ (ð â ((ðī âĪs ðĩ â§ ðĩ â ðī) â ðī <s ðĩ)) |
19 | 11, 18 | impbid 211 | 1 âĒ (ð â (ðī <s ðĩ â (ðī âĪs ðĩ â§ ðĩ â ðī))) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 â§ wa 395 âĻ wo 846 = wceq 1534 â wcel 2099 â wne 2936 class class class wbr 5143 No csur 27567 <s cslt 27568 âĪs csle 27671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-1o 8481 df-2o 8482 df-no 27570 df-slt 27571 df-sle 27672 |
This theorem is referenced by: nnsgt0 28201 |
Copyright terms: Public domain | W3C validator |