Theorem disjor 5077

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 5063	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
2		ralcom4 3255	. . 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 4305	. . . . . . . . . 10 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶))
6		elin 3921	. . . . . . . . . . 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 3075	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
14		ralcom4 3255	. . . . 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 3075	. . 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 2814	. . . . 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 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 3044  ∃*wrmo 3344   ∩ cin 3904  ∅c0 4286  Disj wdisj 5062

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 2701

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 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rmo 3345  df-v 3440  df-dif 3908  df-in 3912  df-nul 4287  df-disj 5063

This theorem is referenced by:  disjors  5078  disjord  5084  disjiunb  5085  disjxiun  5092  disjxun  5093  otsndisj  5466  qsdisj2  8729  s3sndisj  14892  cshwsdisj  17028  dyadmbl  25517  numedglnl  29107  clwwlknondisj  30073  2wspmdisj  30299  disjnf  32532  disjorsf  32542  poimirlem26  37625  mblfinlem2  37637  grpods  42167  ndisj2  45029  nnfoctbdjlem  46437  iundjiun  46442  otiunsndisjX  47264