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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 19702 | . 2 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ 𝐵) |
4 | 3 | ssriv 3920 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ⊆ wss 3881 ‘cfv 6398 Basecbs 16785 Unitcui 19682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-ov 7235 df-dvdsr 19684 df-unit 19685 |
This theorem is referenced by: unitgrpbas 19709 unitgrpid 19712 unitsubm 19713 invrpropd 19741 issubdrg 19850 fidomndrng 20370 znunithash 20554 dvrcn 23105 nmdvr 23592 nrginvrcnlem 23613 nrginvrcn 23614 dchrelbasd 26144 dchrinvcl 26158 dchrghm 26161 dchr1 26162 dchreq 26163 dchrresb 26164 dchrabs 26165 dchrinv 26166 dchrptlem1 26169 dchrptlem2 26170 dchrpt 26172 dchrsum2 26173 dchrsum 26174 sum2dchr 26179 lgsdchr 26260 rpvmasum2 26417 dvrdir 31230 rdivmuldivd 31231 dvrcan5 31233 elrhmunit 31262 rhmunitinv 31264 idomodle 40753 |
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