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 2179, ax-11 2195, ax-12 2216. (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 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-in 3915	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
2	1	eqeq1i 2771	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} = ∅)
3		eleq1w 2849	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		eleq1w 2849	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
5	3, 4	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
6	5	eqabcbw 2840	. . 3 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} = ∅ ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
7		imnan 405	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
8		noel 4294	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
9	8	nbn 375	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
10	7, 9	bitr2i 279	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
11	10	albii 1852	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
12	2, 6, 11	3bitri 300	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
13		df-ral 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
14	12, 13	bitr4i 281	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082   ∩ cin 3907  ∅c0 4289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-dif 3911  df-in 3915  df-nul 4290

This theorem is used by:  disjr  4414  disj1  4415  disjne  4418  disjord  5103  disjiund  5105  otiunsndisj  5508  dfpo2  6304  onxpdisj  6495  f0rn0  6770  onint  7798  zfreg  9568  kmlem4  10156  fin23lem30  10344  fin23lem31  10345  isf32lem3  10357  fpwwe2  10646  renfdisj  11287  fvinim0ffz  13837  s3iunsndisj  15031  metdsge  25044  sltsdisj  28033  dfpth2  30115  2wspmdisj  30725  subfacp1lem1  35692  disjabso  45725  dvmptfprodlem  46699  stoweidlem26  46781  stoweidlem59  46814  iundjiunlem  47214  otiunsndisjX  48057  upgrimpths  48715