Theorem clneq 1168

Description: A way of showing two classes are not equal.

Assertion

Ref	Expression
clneq	⊢ ((A ∈ C ∧ ¬ B ∈ C) → ¬ A = B)

Proof of Theorem clneq

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . 4 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
2	1	biimpcd 137	. . 3 ⊢ (A ∈ C → (A = B → B ∈ C))
3	2	con3d 87	. 2 ⊢ (A ∈ C → (¬ B ∈ C → ¬ A = B))
4	3	imp 277	1 ⊢ ((A ∈ C ∧ ¬ B ∈ C) → ¬ A = B)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  suc11reg 3456

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099

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