Predicate Logic, Howard Pospesel, Prentice Hall, 2003, Chpt. 14, problem 25, p. 213

The argument:

"A set A is a subset of a set B iff there is no member of A that is not a member of B. The empty set E is a subset of every set, since for any set C there is no member of E that is not a member of C, simply because there is no member of E."

The proof:

1. (x)(Ax ⊃Bx) ≡ ¬ (∃x)(Ax • ¬ Bx)
2. ¬ (∃x)(Ex • ¬ Cx)
3. ¬ (∃x)Ex
4. ∴(x)[Ex ⊃(Ax • Bx • Cx)]
5. * Ex ......... ACP
6. * (x) ¬ Ex ......... 3 QC
7. * ¬ Ex ......... 6 UI
8. * ¬ Ex ∨(Ax • Bx • Cx) ......... 7 Add.
9. * Ax • Bx • Cx ......... 8,5 DS
10. Ex ⊃(Ax • Bx • Cx) ......... 5-9 CP
11. (x)[Ex ⊃(Ax • Bx • Cx)] ......... 10 UG