Theorem disj 4406

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2180, ax-11 2197, 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 4094	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
2	1	eqeq1i 2768	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} = ∅)
3		eleq1w 2847	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		eleq1w 2847	. . . . 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 2838	. . 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 4286	. . . . . 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 3081	. 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 2147  {cab 2741  ∀wral 3080   ∩ cin 4092  ∅c0 4281

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-dif 4050  df-in 4094  df-nul 4282

This theorem is used by:  disjr  4407  disj1  4408  disjne  4411  disjord  5095  disjiund  5097  otiunsndisj  5498  dfpo2  6320  onxpdisj  6511  f0rn0  6792  onint  7831  zfreg  9620  kmlem4  10262  fin23lem30  10450  fin23lem31  10451  isf32lem3  10463  fpwwe2  10758  renfdisj  11399  fvinim0ffz  13954  s3iunsndisj  15153  metdsge  25202  sltsdisj  28222  dfpth2  30347  2wspmdisj  30971  subfacp1lem1  36124  disjabso  46171  dvmptfprodlem  47169  stoweidlem26  47251  stoweidlem59  47284  iundjiunlem  47684  otiunsndisjX  48564  upgrimpths  49222