Theorem 0ov 7449

Description: Operation value of the empty set. (Contributed by AV, 15-May-2021.)

Assertion

Ref	Expression
0ov	⊢ (𝐴∅𝐵) = ∅

Proof of Theorem 0ov

Step	Hyp	Ref	Expression
1		df-ov 7415	. 2 ⊢ (𝐴∅𝐵) = (∅‘⟨𝐴, 𝐵⟩)
2		0fv 6924	. 2 ⊢ (∅‘⟨𝐴, 𝐵⟩) = ∅
3	1, 2	eqtri 2786	1 ⊢ (𝐴∅𝐵) = ∅

Syntax hints:   = wceq 1570  ∅c0 4287  ⟨cop 4596  ‘cfv 6538  (class class class)co 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-dm 5673  df-iota 6494  df-fv 6546  df-ov 7415

This theorem is referenced by:  csbov  7457  2mpo0  7661  el2mpocsbcl  8081  homarcl  18086  oppglsm  19713  iswwlksnon  30180  iswspthsnon  30183  mclsrcl  36031  oppcup3  49964  indthinc  50217  indthincALT  50218  prsthinc  50219  lanrcl  50376  ranrcl  50377  rellan  50378  relran  50379