Theorem relsset 31995

Description: The subset class is a binary relation. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
relsset	|- Rel SSet

Proof of Theorem relsset

Step	Hyp	Ref	Expression
1		df-sset 31963	. . 3 |- SSet = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) )
2		difss 3737	. . 3 |- ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) C_ ( _V X. _V )
3	1, 2	eqsstri 3635	. 2 |- SSet C_ ( _V X. _V )
4		df-rel 5121	. 2 |- ( Rel SSet <-> SSet C_ ( _V X. _V ) )
5	3, 4	mpbir 221	1 |- Rel SSet

Syntax hints:   _Vcvv 3200   \ cdif 3571   C_ wss 3574   _E cep 5028   X. cxp 5112   ran crn 5115   Rel wrel 5119   (x) ctxp 31937   SSetcsset 31939

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-rel 5121  df-sset 31963

This theorem is referenced by:  brsset  31996  idsset  31997