Theorem disjor 5148

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 5134	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
2		ralcom4 3292	. . 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 869	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗))
4		df-or 847	. . . . . . 7 ⊢ (((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗) ↔ (¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗))
5		neq0 4375	. . . . . . . . . 10 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶))
6		elin 3992	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
7	6	exbii 1846	. . . . . . . . . 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 1941	. . . . . . . 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 3099	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
14		ralcom4 3292	. . . . 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 3099	. . 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 2830	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
19	18	rmo4 3752	. . . 4 ⊢ (∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
20	19	albii 1817	. . 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 846  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃*wrmo 3387   ∩ cin 3975  ∅c0 4352  Disj wdisj 5133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rmo 3388  df-v 3490  df-dif 3979  df-in 3983  df-nul 4353  df-disj 5134

This theorem is referenced by:  disjors  5149  disjord  5155  disjiunb  5156  disjxiun  5163  disjxun  5164  otsndisj  5538  qsdisj2  8853  s3sndisj  15016  cshwsdisj  17146  dyadmbl  25654  numedglnl  29179  clwwlknondisj  30143  2wspmdisj  30369  disjnf  32592  disjorsf  32602  poimirlem26  37606  mblfinlem2  37618  grpods  42151  ndisj2  44953  nnfoctbdjlem  46376  iundjiun  46381  otiunsndisjX  47194