Theorem 0ov 7400

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 7366	. 2 ⊢ (𝐴∅𝐵) = (∅‘⟨𝐴, 𝐵⟩)
2		0fv 6875	. 2 ⊢ (∅‘⟨𝐴, 𝐵⟩) = ∅
3	1, 2	eqtri 2763	1 ⊢ (𝐴∅𝐵) = ∅

Syntax hints:   = wceq 1547  ∅c0 4268  ⟨cop 4568  ‘cfv 6492  (class class class)co 7363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-dm 5635  df-iota 6448  df-fv 6500  df-ov 7366

This theorem is referenced by:  csbov  7408  2mpo0  7612  el2mpocsbcl  8031  homarcl  17993  oppglsm  19615  iswwlksnon  29946  iswspthsnon  29949  mclsrcl  35796  oppcup3  49706  indthinc  49959  indthincALT  49960  prsthinc  49961  lanrcl  50118  ranrcl  50119  rellan  50120  relran  50121