Theorem cvbr2

Description: Binary relation expressing B covers A . Definition of covers in Kalmbach p. 15. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvbr2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∧ ∀ 𝑥 ∈ Cℋ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cvbr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∧ ¬ ∃ 𝑥 ∈ Cℋ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) ) ) )
2		iman	⊢ ( ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ↔ ¬ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ¬ 𝑥 = 𝐵 ) )
3		anass	⊢ ( ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ¬ 𝑥 = 𝐵 ) ↔ ( 𝐴 ⊊ 𝑥 ∧ ( 𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵 ) ) )
4		dfpss2	⊢ ( 𝑥 ⊊ 𝐵 ↔ ( 𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵 ) )
5	4	anbi2i	⊢ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) ↔ ( 𝐴 ⊊ 𝑥 ∧ ( 𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵 ) ) )
6	3 5	bitr4i	⊢ ( ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ∧ ¬ 𝑥 = 𝐵 ) ↔ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) )
7	2 6	xchbinx	⊢ ( ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ↔ ¬ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) )
8	7	ralbii	⊢ ( ∀ 𝑥 ∈ Cℋ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ↔ ∀ 𝑥 ∈ Cℋ ¬ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) )
9		ralnex	⊢ ( ∀ 𝑥 ∈ Cℋ ¬ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) ↔ ¬ ∃ 𝑥 ∈ Cℋ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) )
10	8 9	bitri	⊢ ( ∀ 𝑥 ∈ Cℋ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ↔ ¬ ∃ 𝑥 ∈ Cℋ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) )
11	10	anbi2i	⊢ ( ( 𝐴 ⊊ 𝐵 ∧ ∀ 𝑥 ∈ Cℋ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ) ↔ ( 𝐴 ⊊ 𝐵 ∧ ¬ ∃ 𝑥 ∈ Cℋ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) ) )
12	1 11	bitr4di	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∧ ∀ 𝑥 ∈ Cℋ ( ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 = 𝐵 ) ) ) )