Theorem idssen 7015

Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
idssen	|- _I C_ ~~

Proof of Theorem idssen

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

Step	Hyp	Ref	Expression
1		reli 4883	. 2 |- Rel _I
2		vex 2815	. . . . 5 |- y e. _V
3	2	ideq 4906	. . . 4 |- ( x _I y <-> x = y )
4		vex 2815	. . . . 5 |- x e. _V
5		eqeng 7004	. . . . 5 |- ( x e. _V -> ( x = y -> x ~~ y ) )
6	4, 5	ax-mp 5	. . . 4 |- ( x = y -> x ~~ y )
7	3, 6	sylbi 121	. . 3 |- ( x _I y -> x ~~ y )
8		df-br 4109	. . 3 |- ( x _I y <-> <. x , y >. e. _I )
9		df-br 4109	. . 3 |- ( x ~~ y <-> <. x , y >. e. ~~ )
10	7, 8, 9	3imtr3i 200	. 2 |- ( <. x , y >. e. _I -> <. x , y >. e. ~~ )
11	1, 10	relssi 4840	1 |- _I C_ ~~

Syntax hints:   -> wi 4   e. wcel 2203   _Vcvv 2812   C_ wss 3210   <.cop 3691   class class class wbr 4108   _I cid 4408   ~~ cen 6972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-en 6975

This theorem is referenced by: (None)