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Theorem inssddif 3287
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 3266 . . 3 (𝐴𝐵) ⊆ 𝐴
2 ssddif 3280 . . 3 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))))
31, 2mpbi 144 . 2 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 3283 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
54difeq2i 3161 . 2 (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))) = (𝐴 ∖ (𝐴𝐵))
63, 5sseqtri 3101 1 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  cdif 3038  cin 3040  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-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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