Theorem idssen 6945

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 4857	. 2 |- Rel _I
2		vex 2803	. . . . 5 |- y e. _V
3	2	ideq 4880	. . . 4 |- ( x _I y <-> x = y )
4		vex 2803	. . . . 5 |- x e. _V
5		eqeng 6934	. . . . 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 4087	. . 3 |- ( x _I y <-> <. x , y >. e. _I )
9		df-br 4087	. . 3 |- ( x ~~ y <-> <. x , y >. e. ~~ )
10	7, 8, 9	3imtr3i 200	. 2 |- ( <. x , y >. e. _I -> <. x , y >. e. ~~ )
11	1, 10	relssi 4815	1 |- _I C_ ~~

Syntax hints:   -> wi 4   e. wcel 2200   _Vcvv 2800   C_ wss 3198   <.cop 3670   class class class wbr 4086   _I cid 4383   ~~ cen 6902

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-en 6905

This theorem is referenced by: (None)