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Mirrors > Home > MPE Home > Th. List > grpn0 | Structured version Visualization version GIF version |
Description: A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
grpn0 | ⊢ (𝐺 ∈ Grp → 𝐺 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | grpbn0 18126 | . 2 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ≠ ∅) |
3 | fveq2 6664 | . . . 4 ⊢ (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅)) | |
4 | base0 16530 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | syl6eqr 2874 | . . 3 ⊢ (𝐺 = ∅ → (Base‘𝐺) = ∅) |
6 | 5 | necon3i 3048 | . 2 ⊢ ((Base‘𝐺) ≠ ∅ → 𝐺 ≠ ∅) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 ‘cfv 6349 Basecbs 16477 Grpcgrp 18097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-riota 7108 df-ov 7153 df-slot 16481 df-base 16483 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 |
This theorem is referenced by: lactghmga 18527 |
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