Theorem relsset 36237

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 36205	. . 3 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2		difss 4090	. . 3 ⊢ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ⊆ (V × V)
3	1, 2	eqsstri 3983	. 2 ⊢ SSet ⊆ (V × V)
4		df-rel 5655	. 2 ⊢ (Rel SSet ↔ SSet ⊆ (V × V))
5	3, 4	mpbir 233	1 ⊢ Rel SSet

Syntax hints:  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905   E cep 5547   × cxp 5646  ran crn 5649  Rel wrel 5653   ⊗ ctxp 36179   SSet csset 36181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-ss 3922  df-rel 5655  df-sset 36205

This theorem is referenced by:  brsset  36238  idsset  36239