Theorem reldisj 4366

Description: Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid ax-12 2175. (Revised by Gino Giotto, 28-Jun-2024.)

Assertion

Ref	Expression
reldisj	⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))

Proof of Theorem reldisj

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3886	. . . 4 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		eleq1w 2820	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3		eleq1w 2820	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶))
4	2, 3	imbi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐶)))
5	4	spw 2042	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
6		pm5.44 546	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))))
7		eldif 3876	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))
8	7	imbi2i 339	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)))
9	6, 8	bitr4di 292	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵))))
10	5, 9	syl 17	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵))))
11	1, 10	sylbi 220	. . 3 ⊢ (𝐴 ⊆ 𝐶 → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵))))
12	11	albidv 1928	. 2 ⊢ (𝐴 ⊆ 𝐶 → (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵))))
13		disj1 4365	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
14		dfss2 3886	. 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))
15	12, 13, 14	3bitr4g 317	1 ⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543   ∈ wcel 2110   ∖ cdif 3863   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238

This theorem is referenced by:  disj2  4372  ssdifsn  4701  oacomf1olem  8292  domdifsn  8728  elfiun  9046  cantnfp1lem3  9295  ssxr  10902  structcnvcnv  16706  fidomndrng  20345  elcls  21970  ist1-2  22244  nrmsep2  22253  nrmsep  22254  isnrm3  22256  isreg2  22274  hauscmplem  22303  connsub  22318  iunconnlem  22324  llycmpkgen2  22447  hausdiag  22542  trfil3  22785  isufil2  22805  filufint  22817  blcld  23403  i1fima2  24576  i1fd  24578  nbgrssvwo2  27450  pliguhgr  28567  symgcom2  31072  inunissunidif  35283  poimirlem15  35529  itg2addnclem2  35566  ntrk0kbimka  41326  ntrneicls11  41377  gneispace  41421  opndisj  45869  seposep  45892