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Theorem inssddif 3206
Description: Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
Assertion
Ref Expression
inssddif (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴𝐵))

Proof of Theorem inssddif
StepHypRef Expression
1 inss1 3187 . . 3 (𝐴𝐵) ⊆ 𝐴
2 ssddif 3199 . . 3 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))))
31, 2mpbi 143 . 2 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 3202 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
54difeq2i 3088 . 2 (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))) = (𝐴 ∖ (𝐴𝐵))
63, 5sseqtri 3032 1 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  cdif 2971  cin 2973  wss 2974
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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