Theorem ssddif 3356

Description: Double complement and subset. Similar to ddifss 3360 but inside a class 𝐵 instead of the universal class V. In classical logic the subset operation on the right hand side could be an equality (that is, 𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴). (Contributed by Jim Kingdon, 24-Jul-2018.)

Assertion

Ref	Expression
ssddif	⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴)))

Proof of Theorem ssddif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancr 319	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
2		simpr 109	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐴)
3	2	con2i 617	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	anim2i 340	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
5		eldif 3125	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)))
6		eldif 3125	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
7	6	notbii 658	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
8	7	anbi2i 453	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
9	5, 8	bitri 183	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
10	4, 9	sylibr 133	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)))
11	1, 10	syl6 33	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
12		eldifi 3244	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) → 𝑥 ∈ 𝐵)
13	12	imim2i 12	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
14	11, 13	impbii 125	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
15	14	albii 1458	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
16		dfss2 3131	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
17		dfss2 3131	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
18	15, 16, 17	3bitr4i 211	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   ∈ wcel 2136   ∖ cdif 3113   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129

This theorem is referenced by:  ddifss  3360  inssddif  3363