Theorem disjorf 29392

Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
disjorf.1	⊢ Ⅎ𝑖𝐴
disjorf.2	⊢ Ⅎ𝑗𝐴
disjorf.3	⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)

Assertion

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

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

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

Proof of Theorem disjorf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 4621	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
2		ralcom4 3224	. . 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 402	. . . . . . 7 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗))
4		df-or 385	. . . . . . 7 ⊢ (((𝐵 ∩ 𝐶) = ∅ ∨ 𝑖 = 𝑗) ↔ (¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗))
5		neq0 3930	. . . . . . . . . 10 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶))
6		elin 3796	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
7	6	exbii 1774	. . . . . . . . . 10 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	5, 7	bitri 264	. . . . . . . . 9 ⊢ (¬ (𝐵 ∩ 𝐶) = ∅ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
9	8	imbi1i 339	. . . . . . . 8 ⊢ ((¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
10		19.23v 1902	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
11	9, 10	bitr4i 267	. . . . . . 7 ⊢ ((¬ (𝐵 ∩ 𝐶) = ∅ → 𝑖 = 𝑗) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
12	3, 4, 11	3bitri 286	. . . . . 6 ⊢ ((𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
13	12	ralbii 2980	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
14		ralcom4 3224	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
15	13, 14	bitri 264	. . . 4 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
16	15	ralbii 2980	. . 3 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑖 ∈ 𝐴 ∀𝑥∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
17		disjorf.1	. . . . 5 ⊢ Ⅎ𝑖𝐴
18		disjorf.2	. . . . 5 ⊢ Ⅎ𝑗𝐴
19		nfv 1843	. . . . 5 ⊢ Ⅎ𝑖 𝑥 ∈ 𝐶
20		disjorf.3	. . . . . 6 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)
21	20	eleq2d 2687	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
22	17, 18, 19, 21	rmo4f 29337	. . . 4 ⊢ (∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
23	22	albii 1747	. . 3 ⊢ (∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) → 𝑖 = 𝑗))
24	2, 16, 23	3bitr4i 292	. 2 ⊢ (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∀𝑥∃*𝑖 ∈ 𝐴 𝑥 ∈ 𝐵)
25	1, 24	bitr4i 267	1 ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  ∃*wrmo 2915   ∩ cin 3573  ∅c0 3915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by: (None)