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| Description: 'Less than' is a strict ordering. Note: do not shorten this with ltsor 5233, and do not use ltsor 5233 in complex number proofs, in order to maintain a portable derivation of all complex number proofs directly from postulates. |
| Ref | Expression |
|---|---|
| ltso |
    |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | axlttri 5475 |
. . . 4
           | |
| 2 | 1 | 3adant3 797 |
. . 3
 |
| 3 | axlttrn 5476 |
. . 3
| |
| 4 | 2, 3 | jca 288 |
. 2
|
| 5 | 4 | so 2855 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wn 2
wi 3
wb 146
wo 222
wa 223
w3a 773
wceq 953
wcel 955
class class class wbr 2609 wor 2830 cr 5205 clt 5458 |
| This theorem is referenced by: lttri2t 5485 lttri3t 5486 ltnrt 5503 lbinfm 5995 suprcl 6002 suprub 6003 suprlub 6004 suprnub 6005 suprcli 6008 suprlubi 6010 suprnubi 6011 infmsup 6015 supxrre 6030 sqrval 6601 sqr0 6602 sqrlem7 6609 sqrlem8 6610 sqrlem13 6615 sqrlem18 6620 caucvg3a 7100 cvgcmp3c 7122 erelem5 7265 erelem6 7266 ele3lem 7268 ege2le3lem1 7269 ege2le3lem2 7271 metxpdval 7769 metxp 7774 xplmi 7907 xplmi2 7908 xplm 7909 xpcn 7910 oprcn 7911 bopcnlem3 7917 bopcn 7919 projlem9 9110 projlem13 9114 projlem15 9116 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 959 ax-gen 960 ax-8 961 ax-9 962 ax-10 963 ax-11 964 ax-12 965 ax-13 966 ax-14 967 ax-17 968 ax-4 970 ax-5o 972 ax-6o 975 ax-9o 1119 ax-10o 1136 ax-16 1206 ax-11o 1213 ax-ext 1452 ax-rep 2683 ax-sep 2693 ax-nul 2700 ax-pow 2732 ax-pr 2769 ax-un 2857 ax-inf2 4597 |
| This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 774 df-3an 775 df-ex 978 df-sb 1168 df-eu 1375 df-mo 1376 df-clab 1457 df-cleq 1462 df-clel 1465 df-ne 1579 df-nel 1580 df-ral 1641 df-rex 1642 df-reu 1643 df-rab 1644 df-v 1803 df-sbc 1932 df-csb 1992 df-dif 2039 df-un 2040 df-in 2041 df-ss 2043 df-pss 2045 df-nul 2271 df-if 2352 df-pw 2392 df-sn 2402 df-pr 2403 df-tp 2405 df-op 2406 df-uni 2494 df-int 2524 df-iun 2558 df-br 2610 df-opab 2657 df-tr 2671 df-eprel 2821 df-id 2824 df-po 2831 df-so 2841 df-fr 2907 df-we 2924 df-ord 2941 df-on 2942 df-lim 2943 df-suc 2944 df-om 3122 df-xp 3174 df-rel 3175 df-cnv 3176 df-co 3177 df-dm 3178 df-rn 3179 df-res 3180 df-ima 3181 df-fun 3182 df-fn 3183 df-f 3184 df-f1 3185 df-fo 3186 df-f1o 3187 df-fv 3188 df-rdg 3917 df-opr 3950 df-oprab 3951 df-1st 4063 df-2nd 4064 df-1o 4117 df-oadd 4119 df-omul 4120 df-er 4245 df-ec 4247 df-qs 4250 df-en 4351 df-dom 4352 df-sdom 4353 df-ni 4972 df-pli 4973 df-mi 4974 df-lti 4975 df-plpq 5007 df-mpq 5008 df-enq 5009 df-nq 5010 df-plq 5011 df-mq 5012 df-rq 5013 df-ltq 5014 df-1q 5015 df-np 5058 df-1p 5059 df-plp 5060 df-ltp 5062 df-enr 5138 df-nr 5139 df-ltr 5142 df-0r 5143 df-c 5212 df-r 5216 df-lt 5219 df-pnf 5459 df-mnf 5460 df-xr 5461 df-ltxr 5462 |