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Theorem difdif2ss 3375
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 3354 . . . 4 (𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2 unss2 3289 . . . 4 ((𝐴𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
31, 2ax-mp 5 . . 3 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4 difindiss 3372 . . 3 ((𝐴𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
53, 4sstri 3147 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6 invdif 3360 . . . 4 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
76eqcomi 2168 . . 3 (𝐵𝐶) = (𝐵 ∩ (V ∖ 𝐶))
87difeq2i 3233 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
95, 8sseqtrri 3173 1 ((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  Vcvv 2722  cdif 3109  cun 3110  cin 3111  wss 3112
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rab 2451  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125
This theorem is referenced by: (None)
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