Theorem disj 4355

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2142, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 28-Jun-2024.)

Assertion

Ref	Expression
disj	⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-in 3888	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2	1	eqeq1i 2803	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅)
3		dfcleq 2792	. . . . 5 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}))
4		df-clab 2777	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ [𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
5		sb6 2090	. . . . . . . 8 ⊢ ([𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
6		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
7		eleq1w 2872	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8	7	biimpd 232	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))
9		eleq1w 2872	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
10	9	biimpd 232	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 → 𝑥 ∈ 𝐵))
11	8, 10	anim12d 611	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
12	6, 11	embantd 59	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
13	12	spimvw 2002	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14		eleq1a 2885	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐴))
15		eleq1a 2885	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐵))
16	14, 15	anim12ii 620	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
17	16	alrimiv 1928	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
18	13, 17	impbii 212	. . . . . . . 8 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
19	4, 5, 18	3bitri 300	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
20	19	bibi2i 341	. . . . . 6 ⊢ ((𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}) ↔ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
21	20	albii 1821	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}) ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
22	3, 21	bitri 278	. . . 4 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
23		eqcom 2805	. . . 4 ⊢ ({𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅ ↔ ∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)})
24		bicom 225	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
25	24	albii 1821	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
26	22, 23, 25	3bitr4i 306	. . 3 ⊢ ({𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅ ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
27		imnan 403	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
28		noel 4247	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
29	28	nbn 376	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
30	27, 29	bitr2i 279	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
31	30	albii 1821	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
32	2, 26, 31	3bitri 300	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
33		df-ral 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
34	32, 33	bitr4i 281	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  [wsb 2069   ∈ wcel 2111  {cab 2776  ∀wral 3106   ∩ cin 3880  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-dif 3884  df-in 3888  df-nul 4244

This theorem is referenced by:  disjr  4357  disj1  4358  disjne  4362  disjord  5018  disjiund  5020  otiunsndisj  5375  onxpdisj  6278  f0rn0  6538  onint  7490  zfreg  9043  kmlem4  9564  fin23lem30  9753  fin23lem31  9754  isf32lem3  9766  fpwwe2  10054  renfdisj  10690  fvinim0ffz  13151  s3iunsndisj  14319  metdsge  23454  2wspmdisj  28122  subfacp1lem1  32539  dfpo2  33104  dvmptfprodlem  42586  stoweidlem26  42668  stoweidlem59  42701  iundjiunlem  43098  otiunsndisjX  43835