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Mirrors > Home > HSE Home > Th. List > chub1i | Structured version Visualization version GIF version |
Description: Cℋ join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chub1i | ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | chshii 29585 | . 2 ⊢ 𝐴 ∈ Sℋ |
3 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | chshii 29585 | . 2 ⊢ 𝐵 ∈ Sℋ |
5 | 2, 4 | shub1i 29732 | 1 ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ⊆ wss 3892 (class class class)co 7271 Cℋ cch 29287 ∨ℋ chj 29291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 ax-hilex 29357 ax-hfvadd 29358 ax-hvcom 29359 ax-hvass 29360 ax-hv0cl 29361 ax-hvaddid 29362 ax-hfvmul 29363 ax-hvmulid 29364 ax-hvmulass 29365 ax-hvdistr1 29366 ax-hvdistr2 29367 ax-hvmul0 29368 ax-hfi 29437 ax-his1 29440 ax-his2 29441 ax-his3 29442 ax-his4 29443 ax-hcompl 29560 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-icc 13085 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cn 22376 df-cnp 22377 df-lm 22378 df-haus 22464 df-tx 22711 df-hmeo 22904 df-xms 23471 df-ms 23472 df-tms 23473 df-cau 24418 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-vs 28957 df-nmcv 28958 df-ims 28959 df-dip 29059 df-hnorm 29326 df-hvsub 29329 df-hlim 29330 df-hcau 29331 df-sh 29565 df-ch 29579 df-oc 29610 df-shs 29666 df-chj 29668 |
This theorem is referenced by: chub2i 29828 chlejb1i 29834 chdmm1i 29835 chnlei 29843 chj00i 29845 lejdii 29896 pjoml4i 29945 pjoml5i 29946 pjoml6i 29947 cmj1i 29962 qlaxr3i 29994 mayetes3i 30087 pjclem1 30553 mdslj1i 30677 mdslmd1lem1 30683 mdslmd2i 30688 mdexchi 30693 atabsi 30759 |
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