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Metamath Proof Explorer

Theorem pr0hash2ex

Description: There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020)

		Ref	Expression
	Assertion	pr0hash2ex	$⊢ \|\{\emptyset, \{\emptyset\}\}\| = 2$

Proof

Step	Hyp	Ref	Expression
1		df2o2	$⊢ 2_{𝑜} = \{\emptyset, \{\emptyset\}\}$
2	1	eqcomi	$⊢ \{\emptyset, \{\emptyset\}\} = 2_{𝑜}$
3	2	fveq2i	$⊢ \|\{\emptyset, \{\emptyset\}\}\| = \|2_{𝑜}\|$
4		hash2	$⊢ \|2_{𝑜}\| = 2$
5	3 4	eqtri	$⊢ \|\{\emptyset, \{\emptyset\}\}\| = 2$