Metamath Proof Explorer


Theorem dtru

Description: Given any set (the " y " in the statement), not all sets are equal to it. The same statement without disjoint variable condition is false since it contradicts stdpc6 . The same comments and revision history concerning axiom usage as in exneq apply. (Contributed by NM, 7-Nov-2006) Extract exneq as an intermediate result. (Revised by BJ, 2-Jan-2025)

Ref Expression
Assertion dtru ¬ x x = y

Proof

Step Hyp Ref Expression
1 exneq x ¬ x = y
2 exnal x ¬ x = y ¬ x x = y
3 1 2 mpbi ¬ x x = y