Theorem undifss 3495

Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undifss	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵)

Proof of Theorem undifss

Step	Hyp	Ref	Expression
1		difss 3253	. . . 4 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
2	1	jctr 313	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵))
3		unss 3301	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵)
4	2, 3	sylib 121	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵)
5		ssun1 3290	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴))
6		sstr 3155	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ (𝐵 ∖ 𝐴)) ∧ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
7	5, 6	mpan 422	. 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
8	4, 7	impbii 125	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵)

Syntax hints:   ∧ wa 103   ↔ wb 104   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  difsnss  3726  exmidundif  4192  exmidundifim  4193  undifdcss  6900