![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2735 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | grpbn0 18997 | . 2 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ≠ ∅) |
3 | fveq2 6907 | . . . 4 ⊢ (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅)) | |
4 | base0 17250 | . . . 4 ⊢ ∅ = (Base‘∅) | |
5 | 3, 4 | eqtr4di 2793 | . . 3 ⊢ (𝐺 = ∅ → (Base‘𝐺) = ∅) |
6 | 5 | necon3i 2971 | . 2 ⊢ ((Base‘𝐺) ≠ ∅ → 𝐺 ≠ ∅) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ‘cfv 6563 Basecbs 17245 Grpcgrp 18964 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 |
This theorem is referenced by: lactghmga 19438 |
Copyright terms: Public domain | W3C validator |