Theorem rev0

Description: The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015)

	Ref	Expression
Assertion	rev0	⊢ ( reverse ‘ ∅ ) = ∅

Proof

Step	Hyp	Ref	Expression
1		wrd0	⊢ ∅ ∈ Word V
2		revlen	⊢ ( ∅ ∈ Word V → ( ♯ ‘ ( reverse ‘ ∅ ) ) = ( ♯ ‘ ∅ ) )
3	1 2	ax-mp	⊢ ( ♯ ‘ ( reverse ‘ ∅ ) ) = ( ♯ ‘ ∅ )
4		hash0	⊢ ( ♯ ‘ ∅ ) = 0
5	3 4	eqtri	⊢ ( ♯ ‘ ( reverse ‘ ∅ ) ) = 0
6		fvex	⊢ ( reverse ‘ ∅ ) ∈ V
7		hasheq0	⊢ ( ( reverse ‘ ∅ ) ∈ V → ( ( ♯ ‘ ( reverse ‘ ∅ ) ) = 0 ↔ ( reverse ‘ ∅ ) = ∅ ) )
8	6 7	ax-mp	⊢ ( ( ♯ ‘ ( reverse ‘ ∅ ) ) = 0 ↔ ( reverse ‘ ∅ ) = ∅ )
9	5 8	mpbi	⊢ ( reverse ‘ ∅ ) = ∅