Theorem inundifss 3471

Description: The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
inundifss	⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴

Proof of Theorem inundifss

Step	Hyp	Ref	Expression
1		inss1 3327	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		difss 3233	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	1, 2	unssi 3282	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴

Syntax hints:   ∖ cdif 3099   ∪ cun 3100   ∩ cin 3101   ⊆ wss 3102

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115

This theorem is referenced by:  resasplitss  5348