Theorem disjor 5081

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 5067	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
2		ralcom4 3287	. . 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 881	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗))
4		df-or 859	. . . . . . 7 ⊢ (((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗) ↔ (¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗))
5		neq0 4304	. . . . . . . . . 10 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶))
6		elin 3920	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
7	6	exbii 1867	. . . . . . . . . 10 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	5, 7	bitri 277	. . . . . . . . 9 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
9	8	imbi1i 351	. . . . . . . 8 ⊢ ((¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
10		19.23v 1961	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
11	9, 10	bitr4i 280	. . . . . . 7 ⊢ ((¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
12	3, 4, 11	3bitri 299	. . . . . 6 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
13	12	ralbii 3107	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
14		ralcom4 3287	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
15	13, 14	bitri 277	. . . 4 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
16	15	ralbii 3107	. . 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 2847	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
19	18	rmo4 3692	. . . 4 ⊢ (∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
20	19	albii 1838	. . 3 ⊢ (∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
21	2, 16, 20	3bitr4i 305	. 2 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
22	1, 21	bitr4i 280	1 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  ∃*wrmo 3365   ∩ cin 3903  ∅c0 4285  Disj wdisj 5066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rmo 3366  df-v 3455  df-dif 3907  df-in 3911  df-nul 4286  df-disj 5067

This theorem is referenced by:  disjors  5082  disjord  5088  disjiunb  5089  disjxiun  5096  disjxun  5097  otsndisj  5487  qsdisj2  8772  s3sndisj  14977  cshwsdisj  17117  dyadmbl  25642  numedglnl  29291  clwwlknondisj  30259  2wspmdisj  30485  disjnf  32719  disjorsf  32729  poimirlem26  38109  mblfinlem2  38121  grpods  42775  ndisj2  45595  nnfoctbdjlem  46993  iundjiun  46998  otiunsndisjX  47837