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Theorem undifss 3447
Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undifss (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)

Proof of Theorem undifss
StepHypRef Expression
1 difss 3206 . . . 4 (𝐵𝐴) ⊆ 𝐵
21jctr 313 . . 3 (𝐴𝐵 → (𝐴𝐵 ∧ (𝐵𝐴) ⊆ 𝐵))
3 unss 3254 . . 3 ((𝐴𝐵 ∧ (𝐵𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
42, 3sylib 121 . 2 (𝐴𝐵 → (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
5 ssun1 3243 . . 3 𝐴 ⊆ (𝐴 ∪ (𝐵𝐴))
6 sstr 3109 . . 3 ((𝐴 ⊆ (𝐴 ∪ (𝐵𝐴)) ∧ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵) → 𝐴𝐵)
75, 6mpan 421 . 2 ((𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵𝐴𝐵)
84, 7impbii 125 1 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  cdif 3072  cun 3073  wss 3075
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-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088
This theorem is referenced by:  difsnss  3673  exmidundif  4136  exmidundifim  4137  undifdcss  6818
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