Theorem 0ov 7429

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 7395	. 2 ⊢ (𝐴∅𝐵) = (∅‘⟨𝐴, 𝐵⟩)
2		0fv 6904	. 2 ⊢ (∅‘⟨𝐴, 𝐵⟩) = ∅
3	1, 2	eqtri 2784	1 ⊢ (𝐴∅𝐵) = ∅

Syntax hints:   = wceq 1559  ∅c0 4285  ⟨cop 4587  ‘cfv 6517  (class class class)co 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-dm 5655  df-iota 6473  df-fv 6525  df-ov 7395

This theorem is referenced by:  csbov  7437  2mpo0  7641  el2mpocsbcl  8059  homarcl  18044  oppglsm  19665  iswwlksnon  29999  iswspthsnon  30002  mclsrcl  35875  oppcup3  49794  indthinc  50047  indthincALT  50048  prsthinc  50049  lanrcl  50206  ranrcl  50207  rellan  50208  relran  50209