This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem disjor

Description: Two ways to say that a collection B ( i ) for i e. A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015) (Revised by Mario Carneiro, 14-Nov-2016)

Ref Expression
Hypothesis disjor.1 ⊢ ( 𝑖 = 𝑗 → 𝐵 = 𝐶 )
Assertion disjor ( Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) )

Proof

Step Hyp Ref Expression
1 disjor.1 ⊢ ( 𝑖 = 𝑗 → 𝐵 = 𝐶 )
2 df-disj ⊢ ( Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∃* 𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 )
3 ralcom4 ⊢ ( ∀ 𝑖 ∈ 𝐴 ∀ 𝑥 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) ↔ ∀ 𝑥 ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
4 orcom ⊢ ( ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ( ( 𝐵 ∩ 𝐶 ) = ∅ ∨ 𝑖 = 𝑗 ) )
5 df-or ⊢ ( ( ( 𝐵 ∩ 𝐶 ) = ∅ ∨ 𝑖 = 𝑗 ) ↔ ( ¬ ( 𝐵 ∩ 𝐶 ) = ∅ → 𝑖 = 𝑗 ) )
6 neq0 ⊢ ( ¬ ( 𝐵 ∩ 𝐶 ) = ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) )
7 elin ⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) )
8 7 exbii ⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) )
9 6 8 bitri ⊢ ( ¬ ( 𝐵 ∩ 𝐶 ) = ∅ ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) )
10 9 imbi1i ⊢ ( ( ¬ ( 𝐵 ∩ 𝐶 ) = ∅ → 𝑖 = 𝑗 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
11 19.23v ⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
12 10 11 bitr4i ⊢ ( ( ¬ ( 𝐵 ∩ 𝐶 ) = ∅ → 𝑖 = 𝑗 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
13 4 5 12 3bitri ⊢ ( ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
14 13 ralbii ⊢ ( ∀ 𝑗 ∈ 𝐴 ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∀ 𝑗 ∈ 𝐴 ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
15 ralcom4 ⊢ ( ∀ 𝑗 ∈ 𝐴 ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) ↔ ∀ 𝑥 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
16 14 15 bitri ⊢ ( ∀ 𝑗 ∈ 𝐴 ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∀ 𝑥 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
17 16 ralbii ⊢ ( ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∀ 𝑖 ∈ 𝐴 ∀ 𝑥 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
18 1 eleq2d ⊢ ( 𝑖 = 𝑗 → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶 ) )
19 18 rmo4 ⊢ ( ∃* 𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
20 19 albii ⊢ ( ∀ 𝑥 ∃* 𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) → 𝑖 = 𝑗 ) )
21 3 17 20 3bitr4i ⊢ ( ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 )
22 2 21 bitr4i ⊢ ( Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( 𝑖 = 𝑗 ∨ ( 𝐵 ∩ 𝐶 ) = ∅ ) )