Metamath Proof Explorer


Theorem sdom1

Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014)

Ref Expression
Assertion sdom1 A 1 𝑜 A =

Proof

Step Hyp Ref Expression
1 domnsym 1 𝑜 A ¬ A 1 𝑜
2 1 con2i A 1 𝑜 ¬ 1 𝑜 A
3 0sdom1dom A 1 𝑜 A
4 2 3 sylnibr A 1 𝑜 ¬ A
5 relsdom Rel
6 5 brrelex1i A 1 𝑜 A V
7 0sdomg A V A A
8 7 necon2bbid A V A = ¬ A
9 6 8 syl A 1 𝑜 A = ¬ A
10 4 9 mpbird A 1 𝑜 A =
11 1n0 1 𝑜
12 1oex 1 𝑜 V
13 12 0sdom 1 𝑜 1 𝑜
14 11 13 mpbir 1 𝑜
15 breq1 A = A 1 𝑜 1 𝑜
16 14 15 mpbiri A = A 1 𝑜
17 10 16 impbii A 1 𝑜 A =