Theorem neleqtrrd 2265

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
neleqtrrd.1	|- ( ph -> -. C e. B )
neleqtrrd.2	|- ( ph -> A = B )

Assertion

Ref	Expression
neleqtrrd	|- ( ph -> -. C e. A )

Proof of Theorem neleqtrrd

Step	Hyp	Ref	Expression
1		neleqtrrd.1	. 2 |- ( ph -> -. C e. B )
2		neleqtrrd.2	. . 3 |- ( ph -> A = B )
3	2	eleq2d 2236	. 2 |- ( ph -> ( C e. A <-> C e. B ) )
4	1, 3	mtbird 663	1 |- ( ph -> -. C e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161

This theorem is referenced by:  tfr1onlemsucaccv  6309  tfrcllemsucaccv  6322  zfz1isolemiso  10752