Theorem ec0 38712

Description: The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.)

Assertion

Ref	Expression
ec0	⊢ [𝐴]∅ = ∅

Proof of Theorem ec0

Step	Hyp	Ref	Expression
1		df-ec 8638	. 2 ⊢ [𝐴]∅ = (∅ “ {𝐴})
2		0ima 6037	. 2 ⊢ (∅ “ {𝐴}) = ∅
3	1, 2	eqtri 2760	1 ⊢ [𝐴]∅ = ∅

Syntax hints:   = wceq 1542  ∅c0 4274  {csn 4568   “ cima 5627  [cec 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8638

This theorem is referenced by:  coss0  38904