Theorem oneptri

Description: The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of Schloeder p. 1. (Contributed by RP, 15-Jan-2025)

	Ref	Expression
Assertion	oneptri	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		epsoon	⊢ E Or On
2		sotrieq	⊢ ( ( E Or On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 = 𝐵 ↔ ¬ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) )
3	1 2	mpan	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 = 𝐵 ↔ ¬ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) )
4		xoror	⊢ ( ( ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ⊻ 𝐴 = 𝐵 ) → ( ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ∨ 𝐴 = 𝐵 ) )
5		xorcom	⊢ ( ( ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ⊻ 𝐴 = 𝐵 ) ↔ ( 𝐴 = 𝐵 ⊻ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) )
6		df-xor	⊢ ( ( 𝐴 = 𝐵 ⊻ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) ↔ ¬ ( 𝐴 = 𝐵 ↔ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) )
7		xor3	⊢ ( ¬ ( 𝐴 = 𝐵 ↔ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) ↔ ( 𝐴 = 𝐵 ↔ ¬ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) )
8	5 6 7	3bitrri	⊢ ( ( 𝐴 = 𝐵 ↔ ¬ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) ↔ ( ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ⊻ 𝐴 = 𝐵 ) )
9		df-3or	⊢ ( ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵 ) ↔ ( ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ∨ 𝐴 = 𝐵 ) )
10	4 8 9	3imtr4i	⊢ ( ( 𝐴 = 𝐵 ↔ ¬ ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ) ) → ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵 ) )
11	3 10	syl	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵 ) )