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Theorem 0ngrp 20894
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 2464 . 2  |-  -.  (/)  =/=  (/)
2 rn0 4952 . . . 4  |-  ran  (/)  =  (/)
32eqcomi 2300 . . 3  |-  (/)  =  ran  (/)
43grpon0 20885 . 2  |-  ( (/)  e.  GrpOp  ->  (/)  =/=  (/) )
51, 4mto 167 1  |-  -.  (/)  e.  GrpOp
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1696    =/= wne 2459   (/)c0 3468   ran crn 4706   GrpOpcgr 20869
This theorem is referenced by:  zrdivrng  21115  vsfval  21207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-grpo 20874
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