Theorem tpssg

Description: An unordered triple of elements of a class is a subset of the class. (Contributed by Thierry Arnoux, 2-Nov-2025)

	Ref	Expression
Assertion	tpssg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷 ) ↔ { 𝐴 , 𝐵 , 𝐶 } ⊆ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		df-3an	⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ 𝐷 ) )
2		prssg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ) ↔ { 𝐴 , 𝐵 } ⊆ 𝐷 ) )
3		snssg	⊢ ( 𝐶 ∈ 𝑋 → ( 𝐶 ∈ 𝐷 ↔ { 𝐶 } ⊆ 𝐷 ) )
4	2 3	bi2anan9	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ 𝐶 ∈ 𝑋 ) → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ 𝐷 ) ↔ ( { 𝐴 , 𝐵 } ⊆ 𝐷 ∧ { 𝐶 } ⊆ 𝐷 ) ) )
5		unss	⊢ ( ( { 𝐴 , 𝐵 } ⊆ 𝐷 ∧ { 𝐶 } ⊆ 𝐷 ) ↔ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) ⊆ 𝐷 )
6		df-tp	⊢ { 𝐴 , 𝐵 , 𝐶 } = ( { 𝐴 , 𝐵 } ∪ { 𝐶 } )
7	6	sseq1i	⊢ ( { 𝐴 , 𝐵 , 𝐶 } ⊆ 𝐷 ↔ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) ⊆ 𝐷 )
8	5 7	bitr4i	⊢ ( ( { 𝐴 , 𝐵 } ⊆ 𝐷 ∧ { 𝐶 } ⊆ 𝐷 ) ↔ { 𝐴 , 𝐵 , 𝐶 } ⊆ 𝐷 )
9	4 8	bitrdi	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ 𝐶 ∈ 𝑋 ) → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ 𝐷 ) ↔ { 𝐴 , 𝐵 , 𝐶 } ⊆ 𝐷 ) )
10	1 9	bitrid	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷 ) ↔ { 𝐴 , 𝐵 , 𝐶 } ⊆ 𝐷 ) )
11	10	3impa	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷 ) ↔ { 𝐴 , 𝐵 , 𝐶 } ⊆ 𝐷 ) )