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Theorem difdif2ss 3194
 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 3173 . . . 4 (𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2 unss2 3114 . . . 4 ((𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
31, 2ax-mp 7 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4 difindiss 3191 . . 3 ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
53, 4sstri 2954 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6 invdif 3179 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
76eqcomi 2044 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
87difeq2i 3059 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
95, 8sseqtr4i 2978 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  Vcvv 2557   ∖ cdif 2914   ∪ cun 2915   ∩ cin 2916   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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