Theorem sspsstri

Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004)

	Ref	Expression
Assertion	sspsstri	⊢ ( ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		or32	⊢ ( ( ( 𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴 ) ∨ 𝐴 = 𝐵 ) ↔ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ∨ 𝐵 ⊊ 𝐴 ) )
2		sspss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) )
3		sspss	⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴 ) )
4		eqcom	⊢ ( 𝐵 = 𝐴 ↔ 𝐴 = 𝐵 )
5	4	orbi2i	⊢ ( ( 𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴 ) ↔ ( 𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵 ) )
6	3 5	bitri	⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵 ) )
7	2 6	orbi12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ↔ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ∨ ( 𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵 ) ) )
8		orordir	⊢ ( ( ( 𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴 ) ∨ 𝐴 = 𝐵 ) ↔ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ∨ ( 𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵 ) ) )
9	7 8	bitr4i	⊢ ( ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ↔ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴 ) ∨ 𝐴 = 𝐵 ) )
10		df-3or	⊢ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴 ) ↔ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ∨ 𝐵 ⊊ 𝐴 ) )
11	1 9 10	3bitr4i	⊢ ( ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴 ) )