Theorem neleqtrd 2887

Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
neleqtrd.1	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
neleqtrd.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
neleqtrd	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

Proof of Theorem neleqtrd

Step	Hyp	Ref	Expression
1		neleqtrd.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
2		neleqtrd.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq2d 2851	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
4	1, 3	mtbid 327	1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-clel 2840

This theorem is used by:  neleqtrrd  2888  smoord  8358  r1tskina  10782  ofccat  15030  mreexexlem2d  17723  chnccat  18704  opptgdim2  29077  lnssplnglem  29124  lnssplng  29125  acopyeu  29196  tgaaddcpbllem2  29204  prlngex  29256  symquadprlng  29267  dimlssid  34086  dochnel  42225  stoweidlem26  46798  fourierdlem60  46938  fourierdlem61  46939  sge00  47148  sge0sn  47151  sge0split  47181