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Theorem inssddif 3378
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 3357 . . 3 (𝐴𝐵) ⊆ 𝐴
2 ssddif 3371 . . 3 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))))
31, 2mpbi 145 . 2 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 3374 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
54difeq2i 3252 . 2 (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))) = (𝐴 ∖ (𝐴𝐵))
63, 5sseqtri 3191 1 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  cdif 3128  cin 3130  wss 3131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144
This theorem is referenced by: (None)
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