Theorem tpprceq3

Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017)

	Ref	Expression
Assertion	tpprceq3	⊢ ( ¬ ( 𝐶 ∈ V ∧ 𝐶 ≠ 𝐵 ) → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		ianor	⊢ ( ¬ ( 𝐶 ∈ V ∧ 𝐶 ≠ 𝐵 ) ↔ ( ¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵 ) )
2		prprc2	⊢ ( ¬ 𝐶 ∈ V → { 𝐵 , 𝐶 } = { 𝐵 } )
3	2	uneq1d	⊢ ( ¬ 𝐶 ∈ V → ( { 𝐵 , 𝐶 } ∪ { 𝐴 } ) = ( { 𝐵 } ∪ { 𝐴 } ) )
4		tprot	⊢ { 𝐴 , 𝐵 , 𝐶 } = { 𝐵 , 𝐶 , 𝐴 }
5		df-tp	⊢ { 𝐵 , 𝐶 , 𝐴 } = ( { 𝐵 , 𝐶 } ∪ { 𝐴 } )
6	4 5	eqtri	⊢ { 𝐴 , 𝐵 , 𝐶 } = ( { 𝐵 , 𝐶 } ∪ { 𝐴 } )
7		prcom	⊢ { 𝐴 , 𝐵 } = { 𝐵 , 𝐴 }
8		df-pr	⊢ { 𝐵 , 𝐴 } = ( { 𝐵 } ∪ { 𝐴 } )
9	7 8	eqtri	⊢ { 𝐴 , 𝐵 } = ( { 𝐵 } ∪ { 𝐴 } )
10	3 6 9	3eqtr4g	⊢ ( ¬ 𝐶 ∈ V → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )
11		nne	⊢ ( ¬ 𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵 )
12		tppreq3	⊢ ( 𝐵 = 𝐶 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )
13	12	eqcoms	⊢ ( 𝐶 = 𝐵 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )
14	11 13	sylbi	⊢ ( ¬ 𝐶 ≠ 𝐵 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )
15	10 14	jaoi	⊢ ( ( ¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵 ) → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )
16	1 15	sylbi	⊢ ( ¬ ( 𝐶 ∈ V ∧ 𝐶 ≠ 𝐵 ) → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )