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Theorem 0ngrp 25937
Description: The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
0ngrp  |-  -.  (/)  e.  GrpOp

Proof of Theorem 0ngrp
StepHypRef Expression
1 neirr 2624 . 2  |-  -.  (/)  =/=  (/)
2 rn0 5105 . . . 4  |-  ran  (/)  =  (/)
32eqcomi 2435 . . 3  |-  (/)  =  ran  (/)
43grpon0 25928 . 2  |-  ( (/)  e.  GrpOp  ->  (/)  =/=  (/) )
51, 4mto 179 1  |-  -.  (/)  e.  GrpOp
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1872    =/= wne 2614   (/)c0 3761   ran crn 4854   GrpOpcgr 25912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-ov 6308  df-grpo 25917
This theorem is referenced by:  zrdivrng  26158  vsfval  26252
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