Theorem mgm0 17255

Description: Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)

Assertion

Ref	Expression
mgm0	|- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Mgm )

Proof of Theorem mgm0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rzal 4073	. . 3 |- ( ( Base ` M ) = (/) -> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) )
2	1	adantl 482	. 2 |- ( ( M e. V /\ ( Base ` M ) = (/) ) -> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) )
3		eqid 2622	. . . 4 |- ( Base ` M ) = ( Base ` M )
4		eqid 2622	. . . 4 |- ( +g ` M ) = ( +g ` M )
5	3, 4	ismgm 17243	. . 3 |- ( M e. V -> ( M e. Mgm <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
6	5	adantr 481	. 2 |- ( ( M e. V /\ ( Base ` M ) = (/) ) -> ( M e. Mgm <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
7	2, 6	mpbird 247	1 |- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Mgm )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242

This theorem is referenced by:  mgm0b  17256  sgrp0  17291