Theorem disj 4404

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) Avoid ax-10 2147, ax-11 2163, ax-12 2185. (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 3910	. . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
2	1	eqeq1i 2742	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} = ∅)
3		eleq1w 2820	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		eleq1w 2820	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
5	3, 4	anbi12d 633	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
6	5	eqabcbw 2811	. . 3 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} = ∅ ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
7		imnan 399	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
8		noel 4292	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
9	8	nbn 372	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅))
10	7, 9	bitr2i 276	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
11	10	albii 1821	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
12	2, 6, 11	3bitri 297	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
13		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
14	12, 13	bitr4i 278	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052   ∩ cin 3902  ∅c0 4287

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-dif 3906  df-in 3910  df-nul 4288

This theorem is referenced by:  disjr  4405  disj1  4406  disjne  4409  disjord  5089  disjiund  5091  otiunsndisj  5476  dfpo2  6262  onxpdisj  6452  f0rn0  6727  onint  7745  zfreg  9513  kmlem4  10076  fin23lem30  10264  fin23lem31  10265  isf32lem3  10277  fpwwe2  10566  renfdisj  11204  fvinim0ffz  13717  s3iunsndisj  14903  metdsge  24806  sltsdisj  27811  dfpth2  29814  2wspmdisj  30424  subfacp1lem1  35392  disjabso  45325  dvmptfprodlem  46296  stoweidlem26  46378  stoweidlem59  46411  iundjiunlem  46811  otiunsndisjX  47633  upgrimpths  48263