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| Mirrors > Home > MPE Home > Th. List > grpbn0 | Structured version Visualization version GIF version | ||
| Description: The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
| Ref | Expression |
|---|---|
| grpbn0.b | ⊢ 𝐵 = (Base‘𝐺) |
| Ref | Expression |
|---|---|
| grpbn0 | ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 3 | 1, 2 | grpidcl 18983 | . 2 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
| 4 | 3 | ne0d 4342 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Grpcgrp 18951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 |
| This theorem is referenced by: grpn0 18989 dfgrp3 19057 issubg2 19159 grpissubg 19164 ghmrn 19247 gexcl3 19605 gexcl2 19607 sylow1lem1 19616 sylow1lem3 19618 sylow1lem5 19620 pgpfi 19623 pgpfi2 19624 sylow2blem3 19640 slwhash 19642 fislw 19643 gexex 19871 lt6abl 19913 ablfac1lem 20088 ablfac1b 20090 ablfac1c 20091 ablfac1eu 20093 pgpfac1lem2 20095 pgpfac1lem3a 20096 ablfaclem3 20107 dvdsr02 20372 0ringnnzr 20525 lmodbn0 20869 lmodsn0 20872 rmodislmodlem 20927 rmodislmod 20928 islss3 20957 rnglidl1 21242 isclmp 25130 qustriv 33392 dfacbasgrp 43120 |
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