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Theorem ssddif 3280
Description: Double complement and subset. Similar to ddifss 3284 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 319 . . . . 5 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
2 simpr 109 . . . . . . . 8 ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → ¬ 𝑥𝐴)
32con2i 601 . . . . . . 7 (𝑥𝐴 → ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
43anim2i 339 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐵 ∧ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
5 eldif 3050 . . . . . . 7 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) ↔ (𝑥𝐵 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
6 eldif 3050 . . . . . . . . 9 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
76notbii 642 . . . . . . . 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 3168 . . . . 5 (𝑥 ∈ (𝐵 ∖ (𝐵𝐴)) → 𝑥𝐵)
1312imim2i 12 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))) → (𝑥𝐴𝑥𝐵))
1411, 13impbii 125 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
1514albii 1431 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵 ∖ (𝐵𝐴))))
16 dfss2 3056 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
17 dfss2 3056 . 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 1314  wcel 1465  cdif 3038  wss 3041
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054
This theorem is referenced by:  ddifss  3284  inssddif  3287
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