Theorem refrel 23836

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 23833	. 2 ⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
2	1	relopabiv 5799	1 ⊢ Rel Ref

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896  ∪ cuni 4866  Rel wrel 5656  Refcref 23830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-opab 5167  df-xp 5657  df-rel 5658  df-ref 23833

This theorem is used by:  isref  23837  refbas  23838  refssex  23839  reftr  23842  refun0  23843  locfinref  34523  refssfne  37242