Theorem inssdif0im 3285

Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Assertion

Ref	Expression
inssdif0im	⊢ ((A ∩ B) ⊆ 𝐶 → (A ∩ (B ∖ 𝐶)) = ∅)

Proof of Theorem inssdif0im

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3120	. . . . . 6 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2	1	imbi1i 227	. . . . 5 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) ↔ ((x ∈ A ∧ x ∈ B) → x ∈ 𝐶))
3		imanim 784	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) → x ∈ 𝐶) → ¬ ((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶))
4	2, 3	sylbi 114	. . . 4 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) → ¬ ((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶))
5		eldif 2921	. . . . . 6 ⊢ (x ∈ (B ∖ 𝐶) ↔ (x ∈ B ∧ ¬ x ∈ 𝐶))
6	5	anbi2i 430	. . . . 5 ⊢ ((x ∈ A ∧ x ∈ (B ∖ 𝐶)) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ 𝐶)))
7		elin 3120	. . . . 5 ⊢ (x ∈ (A ∩ (B ∖ 𝐶)) ↔ (x ∈ A ∧ x ∈ (B ∖ 𝐶)))
8		anass 381	. . . . 5 ⊢ (((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ 𝐶)))
9	6, 7, 8	3bitr4ri 202	. . . 4 ⊢ (((x ∈ A ∧ x ∈ B) ∧ ¬ x ∈ 𝐶) ↔ x ∈ (A ∩ (B ∖ 𝐶)))
10	4, 9	sylnib 600	. . 3 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) → ¬ x ∈ (A ∩ (B ∖ 𝐶)))
11	10	alimi 1341	. 2 ⊢ (∀x(x ∈ (A ∩ B) → x ∈ 𝐶) → ∀x ¬ x ∈ (A ∩ (B ∖ 𝐶)))
12		dfss2 2928	. 2 ⊢ ((A ∩ B) ⊆ 𝐶 ↔ ∀x(x ∈ (A ∩ B) → x ∈ 𝐶))
13		eq0 3233	. 2 ⊢ ((A ∩ (B ∖ 𝐶)) = ∅ ↔ ∀x ¬ x ∈ (A ∩ (B ∖ 𝐶)))
14	11, 12, 13	3imtr4i 190	1 ⊢ ((A ∩ B) ⊆ 𝐶 → (A ∩ (B ∖ 𝐶)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390   ∖ cdif 2908   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219

This theorem is referenced by:  disjdif  3290