Theorem inssdif0 4283

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 3897	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	imbi1i 353	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶))
3		iman 405	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) ↔ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
4	2, 3	bitri 278	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
5		eldif 3891	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
6	5	anbi2i 625	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
7		elin 3897	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)))
8		anass 472	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
9	6, 7, 8	3bitr4ri 307	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
10	4, 9	xchbinx 337	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
11	10	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
12		dfss2 3901	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶))
13		eq0 4258	. 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
14	11, 12, 13	3bitr4i 306	1 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244

This theorem is referenced by:  disjdif  4379  inf3lem3  9077  ssfin4  9721  isnrm2  21963  1stccnp  22067  llycmpkgen2  22155  ufileu  22524  fclscf  22630  flimfnfcls  22633  inindif  30287  opnbnd  33786  diophrw  39700  setindtr  39965