Theorem refrel 21311

Description: Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Assertion

Ref	Expression
refrel	|- Rel Ref

Proof of Theorem refrel

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

Step	Hyp	Ref	Expression
1		df-ref 21308	. 2 |- Ref = { <. x , y >. | ( U. y = U. x /\ A. z e. x E. w e. y z C_ w ) }
2	1	relopabi 5245	1 |- Rel Ref

Syntax hints:   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   Rel wrel 5119   Refcref 21305

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-3an 1039  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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-ref 21308

This theorem is referenced by:  isref  21312  refbas  21313  refssex  21314  reftr  21317  refun0  21318  locfinref  29908  refssfne  32353