Theorem neleqtrd 2882

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 2846	. 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 1569   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-clel 2835

This theorem is used by:  neleqtrrd  2883  smoord  8378  r1tskina  10872  ofccat  15129  mreexexlem2d  17826  chnccat  18807  opptgdim2  29228  lnssplnglem  29276  lnssplng  29277  acopyeu  29349  tgaaddcpbllem2  29357  angmgmaddeu1  29386  prlngex  29436  symquadprlng  29447  dimlssid  34312  dochnel  42557  stoweidlem26  47143  fourierdlem60  47283  fourierdlem61  47284  sge00  47493  sge0sn  47496  sge0split  47526