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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 2737 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | grpbn0 18396 | . 2 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ≠ ∅) |
3 | fveq2 6717 | . . . 4 ⊢ (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅)) | |
4 | base0 16765 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | eqtr4di 2796 | . . 3 ⊢ (𝐺 = ∅ → (Base‘𝐺) = ∅) |
6 | 5 | necon3i 2973 | . 2 ⊢ ((Base‘𝐺) ≠ ∅ → 𝐺 ≠ ∅) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∅c0 4237 ‘cfv 6380 Basecbs 16760 Grpcgrp 18365 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-1cn 10787 ax-addcl 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-nn 11831 df-slot 16735 df-ndx 16745 df-base 16761 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 |
This theorem is referenced by: lactghmga 18797 |
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