Theorem difdif2ss 3338

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	⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶))

Proof of Theorem difdif2ss

Step	Hyp	Ref	Expression
1		inssdif 3317	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2		unss2 3252	. . . 4 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4		difindiss 3335	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5	3, 4	sstri 3111	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6		invdif 3323	. . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
7	6	eqcomi 2144	. . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶))
8	7	difeq2i 3196	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
9	5, 8	sseqtrri 3137	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶))

Syntax hints:  Vcvv 2689   ∖ cdif 3073   ∪ cun 3074   ∩ cin 3075   ⊆ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089

This theorem is referenced by: (None)