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| 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 2948 | . 2 ⊢ ¬ ∅ ≠ ∅ | |
| 2 | rn0 5935 | . . . 4 ⊢ ran ∅ = ∅ | |
| 3 | 2 | eqcomi 2745 | . . 3 ⊢ ∅ = ran ∅ | 
| 4 | 3 | grpon0 30522 | . 2 ⊢ (∅ ∈ GrpOp → ∅ ≠ ∅) | 
| 5 | 1, 4 | mto 197 | 1 ⊢ ¬ ∅ ∈ GrpOp | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 ran crn 5685 GrpOpcgr 30509 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-ov 7435 df-grpo 30513 | 
| This theorem is referenced by: vsfval 30653 zrdivrng 37961 | 
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