Theorem disj 4448

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2138, ax-11 2155, ax-12 2172. (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 3956	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2	1	eqeq1i 2738	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅)
3		dfcleq 2726	. . . . 5 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}))
4		df-clab 2711	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ [𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
5		sb6 2089	. . . . . . . 8 ⊢ ([𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
6		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
7		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8	7	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))
9		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
10	9	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 → 𝑥 ∈ 𝐵))
11	8, 10	anim12d 610	. . . . . . . . . . 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 2000	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14		eleq1a 2829	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐴))
15		eleq1a 2829	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐵))
16	14, 15	anim12ii 619	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
17	16	alrimiv 1931	. . . . . . . . 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 297	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
20	19	bibi2i 338	. . . . . 6 ⊢ ((𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}) ↔ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
21	20	albii 1822	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}) ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
22	3, 21	bitri 275	. . . 4 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
23		eqcom 2740	. . . 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 1822	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
26	22, 23, 25	3bitr4i 303	. . 3 ⊢ ({𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅ ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
27		imnan 401	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
28		noel 4331	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
29	28	nbn 373	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
30	27, 29	bitr2i 276	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
31	30	albii 1822	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
32	2, 26, 31	3bitri 297	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
33		df-ral 3063	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
34	32, 33	bitr4i 278	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540   = wceq 1542  [wsb 2068   ∈ wcel 2107  {cab 2710  ∀wral 3062   ∩ cin 3948  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-dif 3952  df-in 3956  df-nul 4324

This theorem is referenced by:  disjr  4450  disj1  4451  disjne  4455  disjord  5137  disjiund  5139  otiunsndisj  5521  dfpo2  6296  onxpdisj  6491  f0rn0  6777  onint  7778  zfreg  9590  kmlem4  10148  fin23lem30  10337  fin23lem31  10338  isf32lem3  10350  fpwwe2  10638  renfdisj  11274  fvinim0ffz  13751  s3iunsndisj  14915  metdsge  24365  ssltdisj  27322  2wspmdisj  29590  subfacp1lem1  34170  dvmptfprodlem  44660  stoweidlem26  44742  stoweidlem59  44775  iundjiunlem  45175  otiunsndisjX  45987