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Theorem ssddif 3216
Description: Double complement and subset. Similar to ddifss 3220 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, 𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 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 314 . . . . 5 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
2 simpr 108 . . . . . . . 8 ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → ¬ 𝑥𝐴)
32con2i 590 . . . . . . 7 (𝑥𝐴 → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
43anim2i 334 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
5 eldif 2993 . . . . . . 7 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
6 eldif 2993 . . . . . . . . 9 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
76notbii 627 . . . . . . . 8 𝑥 ∈ (𝐵𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
87anbi2i 445 . . . . . . 7 ((𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
95, 8bitri 182 . . . . . 6 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
104, 9sylibr 132 . . . . 5 ((𝑥𝐵𝑥𝐴) → 𝑥 ∈ (𝐵 ∖ (𝐵𝐴)))
111, 10syl6 33 . . . 4 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
12 eldifi 3106 . . . . 5 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) → 𝑥𝐵)
1312imim2i 12 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))) → (𝑥𝐴𝑥𝐵))
1411, 13impbii 124 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
1514albii 1400 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
16 dfss2 2999 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
17 dfss2 2999 . 2 (𝐴 ⊆ (𝐵 ∖ (𝐵𝐴)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
1815, 16, 173bitr4i 210 1 (𝐴𝐵𝐴 ⊆ (𝐵 ∖ (𝐵𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1283  wcel 1434  cdif 2981  wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997
This theorem is referenced by:  ddifss  3220  inssddif  3223
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