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Theorem 0ngrp 21189
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 2534 . 2  |-  -.  (/)  =/=  (/)
2 rn0 5039 . . . 4  |-  ran  (/)  =  (/)
32eqcomi 2370 . . 3  |-  (/)  =  ran  (/)
43grpon0 21180 . 2  |-  ( (/)  e.  GrpOp  ->  (/)  =/=  (/) )
51, 4mto 167 1  |-  -.  (/)  e.  GrpOp
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1715    =/= wne 2529   (/)c0 3543   ran crn 4793   GrpOpcgr 21164
This theorem is referenced by:  zrdivrng  21410  vsfval  21504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-ov 5984  df-grpo 21169
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