Theorem inssdif0im 3264

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 3099	. . . . . 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 773	. . . . 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 2900	. . . . . 6 ⊢ (x ∈ (B ∖ 𝐶) ↔ (x ∈ B ∧ ¬ x ∈ 𝐶))
6	5	anbi2i 433	. . . . 5 ⊢ ((x ∈ A ∧ x ∈ (B ∖ 𝐶)) ↔ (x ∈ A ∧ (x ∈ B ∧ ¬ x ∈ 𝐶)))
7		elin 3099	. . . . 5 ⊢ (x ∈ (A ∩ (B ∖ 𝐶)) ↔ (x ∈ A ∧ x ∈ (B ∖ 𝐶)))
8		anass 383	. . . . 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 588	. . 3 ⊢ ((x ∈ (A ∩ B) → x ∈ 𝐶) → ¬ x ∈ (A ∩ (B ∖ 𝐶)))
11	10	alimi 1320	. 2 ⊢ (∀x(x ∈ (A ∩ B) → x ∈ 𝐶) → ∀x ¬ x ∈ (A ∩ (B ∖ 𝐶)))
12		dfss2 2907	. 2 ⊢ ((A ∩ B) ⊆ 𝐶 ↔ ∀x(x ∈ (A ∩ B) → x ∈ 𝐶))
13		eq0 3212	. 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 1224   = wceq 1226   ∈ wcel 1370   ∖ cdif 2887   ∩ cin 2889   ⊆ wss 2890  ∅c0 3197

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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-nul 3198

This theorem is referenced by:  disjdif  3269