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Metamath Proof Explorer

Theorem ecid

Description: A set is equal to its coset under the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995) (Revised by Mario Carneiro, 9-Jul-2014)

		Ref	Expression
	Hypothesis	ecid.1	⊢ 𝐴 ∈ V
	Assertion	ecid	⊢ [ 𝐴 ] ^◡ E = 𝐴

Proof

Step	Hyp	Ref	Expression
1		ecid.1	⊢ 𝐴 ∈ V
2		vex	⊢ 𝑦 ∈ V
3	2 1	elec	⊢ ( 𝑦 ∈ [ 𝐴 ] ^◡ E ↔ 𝐴 ^◡ E 𝑦 )
4	1 2	brcnv	⊢ ( 𝐴 ^◡ E 𝑦 ↔ 𝑦 E 𝐴 )
5	1	epeli	⊢ ( 𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴 )
6	3 4 5	3bitri	⊢ ( 𝑦 ∈ [ 𝐴 ] ^◡ E ↔ 𝑦 ∈ 𝐴 )
7	6	eqriv	⊢ [ 𝐴 ] ^◡ E = 𝐴