Theorem disjnim 3920

Description: If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)

Hypothesis

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

Assertion

Ref	Expression
disjnim	⊢ (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 disjnim

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 3907	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
2		disjnim.1	. . . . . . 7 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)
3	2	eleq2d 2209	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
4	3	rmo4 2877	. . . . 5 ⊢ (∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
5	4	albii 1446	. . . 4 ⊢ (∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
6		ralcom4 2708	. . . 4 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
7	5, 6	bitr4i 186	. . 3 ⊢ (∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
8		ralcom4 2708	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
9		19.23v 1855	. . . . . . . . 9 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
10	9	biimpi 119	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) → (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
11	10	necon3ad 2350	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) → (𝑖 ≠ 𝑗 → ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
12		notm0 3383	. . . . . . . 8 ⊢ (¬ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝐵 ∩ 𝐶) = ∅)
13		elin 3259	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
14	13	exbii 1584	. . . . . . . . 9 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
15	14	notbii 657	. . . . . . . 8 ⊢ (¬ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
16	12, 15	bitr3i 185	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) = ∅ ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
17	11, 16	syl6ibr 161	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) → (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
18	17	ralimi 2495	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) → ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
19	8, 18	sylbir 134	. . . 4 ⊢ (∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) → ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
20	19	ralimi 2495	. . 3 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
21	7, 20	sylbi 120	. 2 ⊢ (∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
22	1, 21	sylbi 120	1 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  ∃*wrmo 2419   ∩ cin 3070  ∅c0 3363  Disj wdisj 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rmo 2424  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364  df-disj 3907

This theorem is referenced by:  disjnims  3921