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Mirrors > Home > HSE Home > Th. List > chub2i | Structured version Visualization version GIF version |
Description: Cℋ join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chub2i | ⊢ 𝐴 ⊆ (𝐵 ∨ℋ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
3 | 1, 2 | chub1i 29244 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵) |
4 | 1, 2 | chjcomi 29243 | . 2 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
5 | 3, 4 | sseqtri 4000 | 1 ⊢ 𝐴 ⊆ (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 ⊆ wss 3933 (class class class)co 7153 Cℋ cch 28704 ∨ℋ chj 28708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-inf2 9101 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 ax-pre-sup 10612 ax-addf 10613 ax-mulf 10614 ax-hilex 28774 ax-hfvadd 28775 ax-hvcom 28776 ax-hvass 28777 ax-hv0cl 28778 ax-hvaddid 28779 ax-hfvmul 28780 ax-hvmulid 28781 ax-hvmulass 28782 ax-hvdistr1 28783 ax-hvdistr2 28784 ax-hvmul0 28785 ax-hfi 28854 ax-his1 28857 ax-his2 28858 ax-his3 28859 ax-his4 28860 ax-hcompl 28977 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-iin 4919 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-of 7406 df-om 7578 df-1st 7686 df-2nd 7687 df-supp 7828 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-oadd 8103 df-er 8286 df-map 8405 df-pm 8406 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-fsupp 8831 df-fi 8872 df-sup 8903 df-inf 8904 df-oi 8971 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-div 11295 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-q 12347 df-rp 12388 df-xneg 12505 df-xadd 12506 df-xmul 12507 df-ioo 12740 df-icc 12743 df-fz 12891 df-fzo 13032 df-seq 13368 df-exp 13428 df-hash 13689 df-cj 14454 df-re 14455 df-im 14456 df-sqrt 14590 df-abs 14591 df-clim 14841 df-sum 15039 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-starv 16576 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-unif 16584 df-hom 16585 df-cco 16586 df-rest 16692 df-topn 16693 df-0g 16711 df-gsum 16712 df-topgen 16713 df-pt 16714 df-prds 16717 df-xrs 16771 df-qtop 16776 df-imas 16777 df-xps 16779 df-mre 16853 df-mrc 16854 df-acs 16856 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-submnd 17953 df-mulg 18221 df-cntz 18443 df-cmn 18904 df-psmet 20533 df-xmet 20534 df-met 20535 df-bl 20536 df-mopn 20537 df-cnfld 20542 df-top 21498 df-topon 21515 df-topsp 21537 df-bases 21550 df-cn 21831 df-cnp 21832 df-lm 21833 df-haus 21919 df-tx 22166 df-hmeo 22359 df-xms 22926 df-ms 22927 df-tms 22928 df-cau 23855 df-grpo 28268 df-gid 28269 df-ginv 28270 df-gdiv 28271 df-ablo 28320 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-vs 28374 df-nmcv 28375 df-ims 28376 df-dip 28476 df-hnorm 28743 df-hvsub 28746 df-hlim 28747 df-hcau 28748 df-sh 28982 df-ch 28996 df-oc 29027 df-shs 29083 df-chj 29085 |
This theorem is referenced by: chlejb1i 29251 chdmm1i 29252 chj00i 29262 chj1i 29264 lejdii 29313 cmcmlem 29366 cmbr4i 29376 cmj2i 29380 qlaxr3i 29411 osumcori 29418 mayetes3i 29504 pjclem1 29970 pjci 29975 mdslj1i 30094 mdslj2i 30095 mdsl1i 30096 mdsl2i 30097 cvmdi 30099 mdslmd1lem1 30100 mdslmd1lem2 30101 mdslmd2i 30105 mdexchi 30110 sumdmdlem2 30194 dmdbr5ati 30197 |
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