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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 29416 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵) |
4 | 1, 2 | chjcomi 29415 | . 2 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
5 | 3, 4 | sseqtri 3923 | 1 ⊢ 𝐴 ⊆ (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ⊆ wss 3853 (class class class)co 7182 Cℋ cch 28876 ∨ℋ chj 28880 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 ax-mulf 10707 ax-hilex 28946 ax-hfvadd 28947 ax-hvcom 28948 ax-hvass 28949 ax-hv0cl 28950 ax-hvaddid 28951 ax-hfvmul 28952 ax-hvmulid 28953 ax-hvmulass 28954 ax-hvdistr1 28955 ax-hvdistr2 28956 ax-hvmul0 28957 ax-hfi 29026 ax-his1 29029 ax-his2 29030 ax-his3 29031 ax-his4 29032 ax-hcompl 29149 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-icc 12840 df-fz 12994 df-fzo 13137 df-seq 13473 df-exp 13534 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-sum 15148 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-hom 16704 df-cco 16705 df-rest 16811 df-topn 16812 df-0g 16830 df-gsum 16831 df-topgen 16832 df-pt 16833 df-prds 16836 df-xrs 16890 df-qtop 16895 df-imas 16896 df-xps 16898 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-mulg 18355 df-cntz 18577 df-cmn 19038 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-cnfld 20230 df-top 21657 df-topon 21674 df-topsp 21696 df-bases 21709 df-cn 21990 df-cnp 21991 df-lm 21992 df-haus 22078 df-tx 22325 df-hmeo 22518 df-xms 23085 df-ms 23086 df-tms 23087 df-cau 24020 df-grpo 28440 df-gid 28441 df-ginv 28442 df-gdiv 28443 df-ablo 28492 df-vc 28506 df-nv 28539 df-va 28542 df-ba 28543 df-sm 28544 df-0v 28545 df-vs 28546 df-nmcv 28547 df-ims 28548 df-dip 28648 df-hnorm 28915 df-hvsub 28918 df-hlim 28919 df-hcau 28920 df-sh 29154 df-ch 29168 df-oc 29199 df-shs 29255 df-chj 29257 |
This theorem is referenced by: chlejb1i 29423 chdmm1i 29424 chj00i 29434 chj1i 29436 lejdii 29485 cmcmlem 29538 cmbr4i 29548 cmj2i 29552 qlaxr3i 29583 osumcori 29590 mayetes3i 29676 pjclem1 30142 pjci 30147 mdslj1i 30266 mdslj2i 30267 mdsl1i 30268 mdsl2i 30269 cvmdi 30271 mdslmd1lem1 30272 mdslmd1lem2 30273 mdslmd2i 30277 mdexchi 30282 sumdmdlem2 30366 dmdbr5ati 30369 |
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