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Metamath Proof Explorer

Theorem neleqtrrd

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) (Proof shortened by Wolf Lammen, 13-Nov-2019)

Ref Expression
Hypotheses neleqtrrd.1 
|- ( ph -> -. C e. B )
neleqtrrd.2 
|- ( ph -> A = B )
Assertion neleqtrrd 
|- ( ph -> -. C e. A )

Proof

Step Hyp Ref Expression
1 neleqtrrd.1 
 |-  ( ph -> -. C e. B )
2 neleqtrrd.2 
 |-  ( ph -> A = B )
3 2 eqcomd 
 |-  ( ph -> B = A )
4 1 3 neleqtrd 
 |-  ( ph -> -. C e. A )