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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 2939 | . 2 ⊢ ¬ ∅ ≠ ∅ | |
2 | rn0 5922 | . . . 4 ⊢ ran ∅ = ∅ | |
3 | 2 | eqcomi 2735 | . . 3 ⊢ ∅ = ran ∅ |
4 | 3 | grpon0 30429 | . 2 ⊢ (∅ ∈ GrpOp → ∅ ≠ ∅) |
5 | 1, 4 | mto 196 | 1 ⊢ ¬ ∅ ∈ GrpOp |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2099 ≠ wne 2930 ∅c0 4322 ran crn 5673 GrpOpcgr 30416 |
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 2697 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-ov 7416 df-grpo 30420 |
This theorem is referenced by: vsfval 30560 zrdivrng 37664 |
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