![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0ngrp | Structured version Visualization version GIF version |
Description: The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0ngrp | ⊢ ¬ ∅ ∈ GrpOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neirr 3008 | . 2 ⊢ ¬ ∅ ≠ ∅ | |
2 | rn0 5610 | . . . 4 ⊢ ran ∅ = ∅ | |
3 | 2 | eqcomi 2834 | . . 3 ⊢ ∅ = ran ∅ |
4 | 3 | grpon0 27901 | . 2 ⊢ (∅ ∈ GrpOp → ∅ ≠ ∅) |
5 | 1, 4 | mto 189 | 1 ⊢ ¬ ∅ ∈ GrpOp |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2164 ≠ wne 2999 ∅c0 4144 ran crn 5343 GrpOpcgr 27888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fo 6129 df-fv 6131 df-ov 6908 df-grpo 27892 |
This theorem is referenced by: vsfval 28032 zrdivrng 34287 |
Copyright terms: Public domain | W3C validator |