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Metamath Proof Explorer

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 ) ) ) )