Theorem disjor 5092

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

Hypothesis

Ref	Expression
disjor.1	⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)

Assertion

Ref	Expression
disjor	⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))

Distinct variable groups:   𝑖,𝑗,𝐴   𝐵,𝑗   𝐶,𝑖

Allowed substitution hints:   𝐵(𝑖)   𝐶(𝑗)

Proof of Theorem disjor

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 5078	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
2		ralcom4 3264	. . 3 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
3		orcom 870	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗))
4		df-or 848	. . . . . . 7 ⊢ (((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗) ↔ (¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗))
5		neq0 4318	. . . . . . . . . 10 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶))
6		elin 3933	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
7	6	exbii 1848	. . . . . . . . . 10 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	5, 7	bitri 275	. . . . . . . . 9 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
9	8	imbi1i 349	. . . . . . . 8 ⊢ ((¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
10		19.23v 1942	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
11	9, 10	bitr4i 278	. . . . . . 7 ⊢ ((¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
12	3, 4, 11	3bitri 297	. . . . . 6 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
13	12	ralbii 3076	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
14		ralcom4 3264	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
15	13, 14	bitri 275	. . . 4 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
16	15	ralbii 3076	. . 3 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑖 ∈ 𝐴 ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
17		disjor.1	. . . . . 6 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)
18	17	eleq2d 2815	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
19	18	rmo4 3704	. . . 4 ⊢ (∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
20	19	albii 1819	. . 3 ⊢ (∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
21	2, 16, 20	3bitr4i 303	. 2 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
22	1, 21	bitr4i 278	1 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃*wrmo 3355   ∩ cin 3916  ∅c0 4299  Disj wdisj 5077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rmo 3356  df-v 3452  df-dif 3920  df-in 3924  df-nul 4300  df-disj 5078

This theorem is referenced by:  disjors  5093  disjord  5099  disjiunb  5100  disjxiun  5107  disjxun  5108  otsndisj  5482  qsdisj2  8771  s3sndisj  14940  cshwsdisj  17076  dyadmbl  25508  numedglnl  29078  clwwlknondisj  30047  2wspmdisj  30273  disjnf  32506  disjorsf  32516  poimirlem26  37647  mblfinlem2  37659  grpods  42189  ndisj2  45052  nnfoctbdjlem  46460  iundjiun  46465  otiunsndisjX  47284