Theorem disj 4407

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2137, ax-11 2154, ax-12 2171. (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 3917	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2	1	eqeq1i 2741	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅)
3		dfcleq 2729	. . . . 5 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}))
4		df-clab 2714	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ [𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
5		sb6 2088	. . . . . . . 8 ⊢ ([𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
6		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
7		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8	7	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))
9		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
10	9	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 → 𝑥 ∈ 𝐵))
11	8, 10	anim12d 609	. . . . . . . . . . 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 1999	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14		eleq1a 2833	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐴))
15		eleq1a 2833	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐵))
16	14, 15	anim12ii 618	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
17	16	alrimiv 1930	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
18	13, 17	impbii 208	. . . . . . . 8 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
19	4, 5, 18	3bitri 296	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
20	19	bibi2i 337	. . . . . 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 274	. . . 4 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
23		eqcom 2743	. . . 4 ⊢ ({𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅ ↔ ∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)})
24		bicom 221	. . . . 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 302	. . 3 ⊢ ({𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅ ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
27		imnan 400	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
28		noel 4290	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
29	28	nbn 372	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
30	27, 29	bitr2i 275	. . . 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 296	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
33		df-ral 3065	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
34	32, 33	bitr4i 277	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541  [wsb 2067   ∈ wcel 2106  {cab 2713  ∀wral 3064   ∩ cin 3909  ∅c0 4282

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-dif 3913  df-in 3917  df-nul 4283

This theorem is referenced by:  disjr  4409  disj1  4410  disjne  4414  disjord  5093  disjiund  5095  otiunsndisj  5477  dfpo2  6248  onxpdisj  6443  f0rn0  6727  onint  7725  zfreg  9531  kmlem4  10089  fin23lem30  10278  fin23lem31  10279  isf32lem3  10291  fpwwe2  10579  renfdisj  11215  fvinim0ffz  13691  s3iunsndisj  14853  metdsge  24212  ssltdisj  27160  2wspmdisj  29281  subfacp1lem1  33773  dvmptfprodlem  44175  stoweidlem26  44257  stoweidlem59  44290  iundjiunlem  44690  otiunsndisjX  45501