Theorem inssdif0 3617

Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
inssdif0	⊢ ((A ∩ B) ⊆ C ↔ (A ∩ (B ∖ C)) = ∅)

Proof of Theorem inssdif0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3219	. . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2	1	imbi1i 315	. . . . 5 ⊢ ((x ∈ (A ∩ B) → x ∈ C) ↔ ((x ∈ A ∧ x ∈ B) → x ∈ C))
3		iman 413	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) → x ∈ C) ↔ ¬ ((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ C))
4	2, 3	bitri 240	. . . 4 ⊢ ((x ∈ (A ∩ B) → x ∈ C) ↔ ¬ ((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ C))
5		eldif 3221	. . . . . 6 ⊢ (x ∈ (B ∖ C) ↔ (x ∈ B ∧ ¬ x ∈ C))
6	5	anbi2i 675	. . . . 5 ⊢ ((x ∈ A ∧ x ∈ (B ∖ C)) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
7		elin 3219	. . . . 5 ⊢ (x ∈ (A ∩ (B ∖ C)) ↔ (x ∈ A ∧ x ∈ (B ∖ C)))
8		anass 630	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ C) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ C)))
9	6, 7, 8	3bitr4ri 269	. . . 4 ⊢ (((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ C) ↔ x ∈ (A ∩ (B ∖ C)))
10	4, 9	xchbinx 301	. . 3 ⊢ ((x ∈ (A ∩ B) → x ∈ C) ↔ ¬ x ∈ (A ∩ (B ∖ C)))
11	10	albii 1566	. 2 ⊢ (∀x(x ∈ (A ∩ B) → x ∈ C) ↔ ∀x ¬ x ∈ (A ∩ (B ∖ C)))
12		dfss2 3262	. 2 ⊢ ((A ∩ B) ⊆ C ↔ ∀x(x ∈ (A ∩ B) → x ∈ C))
13		eq0 3564	. 2 ⊢ ((A ∩ (B ∖ C)) = ∅ ↔ ∀x ¬ x ∈ (A ∩ (B ∖ C)))
14	11, 12, 13	3bitr4i 268	1 ⊢ ((A ∩ B) ⊆ C ↔ (A ∩ (B ∖ C)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by:  disjdif  3622