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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 18482 | . 2 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ 𝐵) |
4 | 3 | ssriv 3572 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊆ wss 3540 ‘cfv 5804 Basecbs 15695 Unitcui 18462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-dvdsr 18464 df-unit 18465 |
This theorem is referenced by: unitgrpbas 18489 unitgrpid 18492 unitsubm 18493 invrpropd 18521 issubdrg 18628 fidomndrng 19128 znunithash 19732 dvrcn 21797 nmdvr 22284 nrginvrcnlem 22305 nrginvrcn 22306 dchrelbasd 24764 dchrinvcl 24778 dchrghm 24781 dchr1 24782 dchreq 24783 dchrresb 24784 dchrabs 24785 dchrinv 24786 dchrptlem1 24789 dchrptlem2 24790 dchrpt 24792 dchrsum2 24793 dchrsum 24794 sum2dchr 24799 lgsdchr 24880 rpvmasum2 25001 dvrdir 29121 rdivmuldivd 29122 dvrcan5 29124 elrhmunit 29151 rhmunitinv 29153 idomodle 36793 |
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