Theorem difdif2ss 3221

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 3200	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶))
2		unss2 3143	. . . 4 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))))
3	1, 2	ax-mp 7	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
4		difindiss 3218	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5	3, 4	sstri 3008	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
6		invdif 3206	. . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
7	6	eqcomi 2085	. . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶))
8	7	difeq2i 3087	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
9	5, 8	sseqtr4i 3032	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶))

Syntax hints:  Vcvv 2601   ∖ cdif 2970   ∪ cun 2971   ∩ cin 2972   ⊆ wss 2973

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by: (None)