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Theorem inssddif 3283
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 3262 . . 3 (𝐴𝐵) ⊆ 𝐴
2 ssddif 3276 . . 3 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))))
31, 2mpbi 144 . 2 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 3279 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
54difeq2i 3157 . 2 (𝐴 ∖ (𝐴 ∖ (𝐴𝐵))) = (𝐴 ∖ (𝐴𝐵))
63, 5sseqtri 3097 1 (𝐴𝐵) ⊆ (𝐴 ∖ (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  cdif 3034  cin 3036  wss 3037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050
This theorem is referenced by: (None)
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