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Mirrors > Home > MPE Home > Th. List > unitss | Structured version Visualization version GIF version |
Description: The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
unitcl.1 | ⊢ 𝐵 = (Base‘𝑅) |
unitcl.2 | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
unitss | ⊢ 𝑈 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitcl.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | unitcl.2 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | 1, 2 | unitcl 20352 | . 2 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ 𝐵) |
4 | 3 | ssriv 3984 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ⊆ wss 3948 ‘cfv 6545 Basecbs 17207 Unitcui 20332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6497 df-fun 6547 df-fv 6553 df-ov 7418 df-dvdsr 20334 df-unit 20335 |
This theorem is referenced by: unitgrpbas 20359 unitgrpid 20362 unitsubm 20363 dvrdir 20389 rdivmuldivd 20390 invrpropd 20395 elrhmunit 20487 rhmunitinv 20488 fidomndrng 20747 issubdrg 20754 imadrhmcl 20771 znunithash 21557 dvrcn 24175 nmdvr 24674 nrginvrcnlem 24695 nrginvrcn 24696 dchrelbasd 27264 dchrinvcl 27278 dchrghm 27281 dchr1 27282 dchreq 27283 dchrresb 27284 dchrabs 27285 dchrinv 27286 dchrptlem1 27289 dchrptlem2 27290 dchrpt 27292 dchrsum2 27293 dchrsum 27294 sum2dchr 27299 lgsdchr 27380 rpvmasum2 27537 dvrcan5 33105 isdrng4 33151 dvdsruassoi 33264 lidlunitel 33303 idomodle 42892 |
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