The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.5E, problem 5(b), p. 563

The following set of sentences is inconsistent: {¬ Fa , ¬ (∃x)(¬ Fx ∨¬ Fx)} and so much needs to be proved. Hint: working only with the assumptions given, we easily derive a contradiction.

1. ¬ Fa / Assumption
2. ¬ (∃x)(¬ Fx ∨¬ Fx) / Assumption
3. (x) ¬ (¬ Fx ∨¬ Fx) / 2QC
4. (x)(Fx • Fx) / 3DeM
5. (x)Fx / 4Taut.
6. Fa / 5UI x/a
7. Fa • ¬ Fa / 6,1Conj.