Theorem ssddif 3438

Description: Double complement and subset. Similar to ddifss 3442 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 321	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
2		simpr 110	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐴)
3	2	con2i 630	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	anim2i 342	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
5		eldif 3206	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)))
6		eldif 3206	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
7	6	notbii 672	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
8	7	anbi2i 457	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
9	5, 8	bitri 184	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
10	4, 9	sylibr 134	. . . . 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 3326	. . . . 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 126	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
15	14	albii 1516	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
16		ssalel 3212	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
17		ssalel 3212	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴))))
18	15, 16, 17	3bitr4i 212	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393   ∈ wcel 2200   ∖ cdif 3194   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210

This theorem is referenced by:  ddifss  3442  inssddif  3445