Theorem relsset 32588

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 32556	. . 3 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2		difss 3960	. . 3 ⊢ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ⊆ (V × V)
3	1, 2	eqsstri 3854	. 2 ⊢ SSet ⊆ (V × V)
4		df-rel 5364	. 2 ⊢ (Rel SSet ↔ SSet ⊆ (V × V))
5	3, 4	mpbir 223	1 ⊢ Rel SSet

Syntax hints:  Vcvv 3398   ∖ cdif 3789   ⊆ wss 3792   E cep 5267   × cxp 5355  ran crn 5358  Rel wrel 5362   ⊗ ctxp 32530   SSet csset 32532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-dif 3795  df-in 3799  df-ss 3806  df-rel 5364  df-sset 32556

This theorem is referenced by:  brsset  32589  idsset  32590