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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 19338 | . 2 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ 𝐵) |
4 | 3 | ssriv 3968 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊆ wss 3933 ‘cfv 6348 Basecbs 16471 Unitcui 19318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-dvdsr 19320 df-unit 19321 |
This theorem is referenced by: unitgrpbas 19345 unitgrpid 19348 unitsubm 19349 invrpropd 19377 issubdrg 19489 fidomndrng 20008 znunithash 20639 dvrcn 22719 nmdvr 23206 nrginvrcnlem 23227 nrginvrcn 23228 dchrelbasd 25742 dchrinvcl 25756 dchrghm 25759 dchr1 25760 dchreq 25761 dchrresb 25762 dchrabs 25763 dchrinv 25764 dchrptlem1 25767 dchrptlem2 25768 dchrpt 25770 dchrsum2 25771 dchrsum 25772 sum2dchr 25777 lgsdchr 25858 rpvmasum2 26015 dvrdir 30788 rdivmuldivd 30789 dvrcan5 30791 elrhmunit 30820 rhmunitinv 30822 idomodle 39674 |
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