Theorem inssdif0im 3369

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 3198	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	imbi1i 237	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶))
3		imanim 826	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
4	2, 3	sylbi 120	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
5		eldif 3022	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
6	5	anbi2i 446	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
7		elin 3198	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∖ 𝐶)))
8		anass 394	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
9	6, 7, 8	3bitr4ri 212	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
10	4, 9	sylnib 639	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
11	10	alimi 1396	. 2 ⊢ (∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶) → ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
12		dfss2 3028	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐶))
13		eq0 3320	. 2 ⊢ ((𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 ∖ 𝐶)))
14	11, 12, 13	3imtr4i 200	1 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1294   = wceq 1296   ∈ wcel 1445   ∖ cdif 3010   ∩ cin 3012   ⊆ wss 3013  ∅c0 3302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303

This theorem is referenced by:  disjdif  3374