Theorem neleqtrrd 2330

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 2301	. 2 |- ( ph -> ( C e. A <-> C e. B ) )
4	1, 3	mtbird 680	1 |- ( ph -> -. C e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 2202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  tfr1onlemsucaccv  6550  tfrcllemsucaccv  6563  zfz1isolemiso  11166  wrdlndm  11196  structiedg0val  15981  1hevtxdg0fi  16248