Metamath Proof Explorer


Theorem 0ring

Description: If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019)

Ref Expression
Hypotheses 0ring.b B = Base R
0ring.0 0 ˙ = 0 R
Assertion 0ring R Ring B = 1 B = 0 ˙

Proof

Step Hyp Ref Expression
1 0ring.b B = Base R
2 0ring.0 0 ˙ = 0 R
3 1 2 ring0cl R Ring 0 ˙ B
4 1 fvexi B V
5 hashen1 B V B = 1 B 1 𝑜
6 4 5 ax-mp B = 1 B 1 𝑜
7 en1eqsn 0 ˙ B B 1 𝑜 B = 0 ˙
8 7 ex 0 ˙ B B 1 𝑜 B = 0 ˙
9 6 8 syl5bi 0 ˙ B B = 1 B = 0 ˙
10 3 9 syl R Ring B = 1 B = 0 ˙
11 10 imp R Ring B = 1 B = 0 ˙