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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 29840 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∨ℋ 𝐵) |
4 | 1, 2 | chjcomi 29839 | . 2 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
5 | 3, 4 | sseqtri 3958 | 1 ⊢ 𝐴 ⊆ (𝐵 ∨ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ⊆ wss 3888 (class class class)co 7284 Cℋ cch 29300 ∨ℋ chj 29304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-inf2 9408 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958 ax-addf 10959 ax-mulf 10960 ax-hilex 29370 ax-hfvadd 29371 ax-hvcom 29372 ax-hvass 29373 ax-hv0cl 29374 ax-hvaddid 29375 ax-hfvmul 29376 ax-hvmulid 29377 ax-hvmulass 29378 ax-hvdistr1 29379 ax-hvdistr2 29380 ax-hvmul0 29381 ax-hfi 29450 ax-his1 29453 ax-his2 29454 ax-his3 29455 ax-his4 29456 ax-hcompl 29573 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 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 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-isom 6446 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-of 7542 df-om 7722 df-1st 7840 df-2nd 7841 df-supp 7987 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-2o 8307 df-er 8507 df-map 8626 df-pm 8627 df-ixp 8695 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-fsupp 9138 df-fi 9179 df-sup 9210 df-inf 9211 df-oi 9278 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-z 12329 df-dec 12447 df-uz 12592 df-q 12698 df-rp 12740 df-xneg 12857 df-xadd 12858 df-xmul 12859 df-ioo 13092 df-icc 13095 df-fz 13249 df-fzo 13392 df-seq 13731 df-exp 13792 df-hash 14054 df-cj 14819 df-re 14820 df-im 14821 df-sqrt 14955 df-abs 14956 df-clim 15206 df-sum 15407 df-struct 16857 df-sets 16874 df-slot 16892 df-ndx 16904 df-base 16922 df-ress 16951 df-plusg 16984 df-mulr 16985 df-starv 16986 df-sca 16987 df-vsca 16988 df-ip 16989 df-tset 16990 df-ple 16991 df-ds 16993 df-unif 16994 df-hom 16995 df-cco 16996 df-rest 17142 df-topn 17143 df-0g 17161 df-gsum 17162 df-topgen 17163 df-pt 17164 df-prds 17167 df-xrs 17222 df-qtop 17227 df-imas 17228 df-xps 17230 df-mre 17304 df-mrc 17305 df-acs 17307 df-mgm 18335 df-sgrp 18384 df-mnd 18395 df-submnd 18440 df-mulg 18710 df-cntz 18932 df-cmn 19397 df-psmet 20598 df-xmet 20599 df-met 20600 df-bl 20601 df-mopn 20602 df-cnfld 20607 df-top 22052 df-topon 22069 df-topsp 22091 df-bases 22105 df-cn 22387 df-cnp 22388 df-lm 22389 df-haus 22475 df-tx 22722 df-hmeo 22915 df-xms 23482 df-ms 23483 df-tms 23484 df-cau 24429 df-grpo 28864 df-gid 28865 df-ginv 28866 df-gdiv 28867 df-ablo 28916 df-vc 28930 df-nv 28963 df-va 28966 df-ba 28967 df-sm 28968 df-0v 28969 df-vs 28970 df-nmcv 28971 df-ims 28972 df-dip 29072 df-hnorm 29339 df-hvsub 29342 df-hlim 29343 df-hcau 29344 df-sh 29578 df-ch 29592 df-oc 29623 df-shs 29679 df-chj 29681 |
This theorem is referenced by: chlejb1i 29847 chdmm1i 29848 chj00i 29858 chj1i 29860 lejdii 29909 cmcmlem 29962 cmbr4i 29972 cmj2i 29976 qlaxr3i 30007 osumcori 30014 mayetes3i 30100 pjclem1 30566 pjci 30571 mdslj1i 30690 mdslj2i 30691 mdsl1i 30692 mdsl2i 30693 cvmdi 30695 mdslmd1lem1 30696 mdslmd1lem2 30697 mdslmd2i 30701 mdexchi 30706 sumdmdlem2 30790 dmdbr5ati 30793 |
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