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Theorem ssddif 3310
 Description: Double complement and subset. Similar to ddifss 3314 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 (𝐴𝐵𝐴 ⊆ (𝐵 ∖ (𝐵𝐴)))

Proof of Theorem ssddif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ancr 319 . . . . 5 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
2 simpr 109 . . . . . . . 8 ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → ¬ 𝑥𝐴)
32con2i 616 . . . . . . 7 (𝑥𝐴 → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
43anim2i 339 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
5 eldif 3080 . . . . . . 7 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
6 eldif 3080 . . . . . . . . 9 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
76notbii 657 . . . . . . . 8 𝑥 ∈ (𝐵𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
87anbi2i 452 . . . . . . 7 ((𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
95, 8bitri 183 . . . . . 6 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
104, 9sylibr 133 . . . . 5 ((𝑥𝐵𝑥𝐴) → 𝑥 ∈ (𝐵 ∖ (𝐵𝐴)))
111, 10syl6 33 . . . 4 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
12 eldifi 3198 . . . . 5 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) → 𝑥𝐵)
1312imim2i 12 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))) → (𝑥𝐴𝑥𝐵))
1411, 13impbii 125 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
1514albii 1446 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
16 dfss2 3086 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
17 dfss2 3086 . 2 (𝐴 ⊆ (𝐵 ∖ (𝐵𝐴)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
1815, 16, 173bitr4i 211 1 (𝐴𝐵𝐴 ⊆ (𝐵 ∖ (𝐵𝐴)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   ∈ wcel 1480   ∖ cdif 3068   ⊆ wss 3071 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084 This theorem is referenced by:  ddifss  3314  inssddif  3317
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