Theorem ttc00

Description: A class has an empty transitive closure iff it is the empty set. (Contributed by Matthew House, 6-Apr-2026)

	Ref	Expression
Assertion	ttc00	Could not format assertion : No typesetting found for |- ( A = (/) <-> TC+ A = (/) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		ttceq	Could not format ( A = (/) -> TC+ A = TC+ (/) ) : No typesetting found for |- ( A = (/) -> TC+ A = TC+ (/) ) with typecode |-
2		ttc0	Could not format TC+ (/) = (/) : No typesetting found for |- TC+ (/) = (/) with typecode |-
3	1 2	eqtrdi	Could not format ( A = (/) -> TC+ A = (/) ) : No typesetting found for |- ( A = (/) -> TC+ A = (/) ) with typecode |-
4		ttcid	Could not format A C_ TC+ A : No typesetting found for |- A C_ TC+ A with typecode |-
5		sseq2	Could not format ( TC+ A = (/) -> ( A C_ TC+ A <-> A C_ (/) ) ) : No typesetting found for |- ( TC+ A = (/) -> ( A C_ TC+ A <-> A C_ (/) ) ) with typecode |-
6	4 5	mpbii	Could not format ( TC+ A = (/) -> A C_ (/) ) : No typesetting found for |- ( TC+ A = (/) -> A C_ (/) ) with typecode |-
7		ss0	$$⊢A\subseteq \varnothing \to A=\varnothing$$
8	6 7	syl	Could not format ( TC+ A = (/) -> A = (/) ) : No typesetting found for |- ( TC+ A = (/) -> A = (/) ) with typecode |-
9	3 8	impbii	Could not format ( A = (/) <-> TC+ A = (/) ) : No typesetting found for |- ( A = (/) <-> TC+ A = (/) ) with typecode |-