ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difdif2ss GIF version

Theorem difdif2ss 3256
Description: Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
Assertion
Ref Expression
difdif2ss ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))

Proof of Theorem difdif2ss
StepHypRef Expression
1 inssdif 3235 . . . 4 (𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2 unss2 3171 . . . 4 ((𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
31, 2ax-mp 7 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4 difindiss 3253 . . 3 ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
53, 4sstri 3034 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6 invdif 3241 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
76eqcomi 2092 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
87difeq2i 3115 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
95, 8sseqtr4i 3059 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  Vcvv 2619  cdif 2996  cun 2997  cin 2998  wss 2999
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator