Theorem disj 4413

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 2178. (Revised by GG, 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 3921	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}
2	1	eqeq1i 2734	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅)
3		dfcleq 2722	. . . . 5 ⊢ (∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}))
4		df-clab 2708	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} ↔ [𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
5		sb6 2086	. . . . . . . 8 ⊢ ([𝑥 / 𝑧](𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
6		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
7		eleq1w 2811	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8	7	biimpd 229	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))
9		eleq1w 2811	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
10	9	biimpd 229	. . . . . . . . . . . 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 1986	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
14		eleq1a 2823	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 = 𝑥 → 𝑧 ∈ 𝐴))
15		eleq1a 2823	. . . . . . . . . . 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 1927	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ∀𝑧(𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
18	13, 17	impbii 209	. . . . . . . 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 337	. . . . . 6 ⊢ ((𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)}) ↔ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
21	20	albii 1819	. . . . 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 2736	. . . 4 ⊢ ({𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = ∅ ↔ ∅ = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)})
24		bicom 222	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
25	24	albii 1819	. . . 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 399	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
28		noel 4301	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
29	28	nbn 372	. . . . 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 1819	. . 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 3045	. 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 206   ∧ wa 395  ∀wal 1538   = wceq 1540  [wsb 2065   ∈ wcel 2109  {cab 2707  ∀wral 3044   ∩ cin 3913  ∅c0 4296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-dif 3917  df-in 3921  df-nul 4297

This theorem is referenced by:  disjr  4414  disj1  4415  disjne  4418  disjord  5096  disjiund  5098  otiunsndisj  5480  dfpo2  6269  onxpdisj  6460  f0rn0  6745  onint  7766  zfreg  9548  kmlem4  10107  fin23lem30  10295  fin23lem31  10296  isf32lem3  10308  fpwwe2  10596  renfdisj  11234  fvinim0ffz  13747  s3iunsndisj  14934  metdsge  24738  ssltdisj  27733  dfpth2  29659  2wspmdisj  30266  subfacp1lem1  35166  disjabso  44965  dvmptfprodlem  45942  stoweidlem26  46024  stoweidlem59  46057  iundjiunlem  46457  otiunsndisjX  47277  upgrimpths  47906