Theorem eqneltri 39246

Description: If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
eqneltri.1	|- A = B
eqneltri.2	|- -. B e. C

Assertion

Ref	Expression
eqneltri	|- -. A e. C

Proof of Theorem eqneltri

Step	Hyp	Ref	Expression
1		eqneltri.2	. 2 |- -. B e. C
2		eqneltri.1	. . 3 |- A = B
3	2	eleq1i 2692	. 2 |- ( A e. C <-> B e. C )
4	1, 3	mtbir 313	1 |- -. A e. C

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  eliuniincex  39292  eliincex  39293  salgencntex  40561