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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 2728 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | grpbn0 18922 | . 2 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ≠ ∅) |
3 | fveq2 6897 | . . . 4 ⊢ (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅)) | |
4 | base0 17184 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | eqtr4di 2786 | . . 3 ⊢ (𝐺 = ∅ → (Base‘𝐺) = ∅) |
6 | 5 | necon3i 2970 | . 2 ⊢ ((Base‘𝐺) ≠ ∅ → 𝐺 ≠ ∅) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 ‘cfv 6548 Basecbs 17179 Grpcgrp 18889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-1cn 11196 ax-addcl 11198 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-nn 12243 df-slot 17150 df-ndx 17162 df-base 17180 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 |
This theorem is referenced by: lactghmga 19359 |
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