Theorem 0ov 7437

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 7403	. 2 ⊢ (𝐴∅𝐵) = (∅‘⟨𝐴, 𝐵⟩)
2		0fv 6912	. 2 ⊢ (∅‘⟨𝐴, 𝐵⟩) = ∅
3	1, 2	eqtri 2788	1 ⊢ (𝐴∅𝐵) = ∅

Syntax hints:   = wceq 1563  ∅c0 4288  ⟨cop 4591  ‘cfv 6525  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-dm 5662  df-iota 6481  df-fv 6533  df-ov 7403

This theorem is referenced by:  csbov  7445  2mpo0  7649  el2mpocsbcl  8068  homarcl  18075  oppglsm  19703  iswwlksnon  30111  iswspthsnon  30114  mclsrcl  35924  oppcup3  49838  indthinc  50091  indthincALT  50092  prsthinc  50093  lanrcl  50250  ranrcl  50251  rellan  50252  relran  50253