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Theorem undifss 3344
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 3110 . . . 4 (𝐵𝐴) ⊆ 𝐵
21jctr 308 . . 3 (𝐴𝐵 → (𝐴𝐵 ∧ (𝐵𝐴) ⊆ 𝐵))
3 unss 3158 . . 3 ((𝐴𝐵 ∧ (𝐵𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
42, 3sylib 120 . 2 (𝐴𝐵 → (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
5 ssun1 3147 . . 3 𝐴 ⊆ (𝐴 ∪ (𝐵𝐴))
6 sstr 3018 . . 3 ((𝐴 ⊆ (𝐴 ∪ (𝐵𝐴)) ∧ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵) → 𝐴𝐵)
75, 6mpan 415 . 2 ((𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵𝐴𝐵)
84, 7impbii 124 1 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  cdif 2981  cun 2982  wss 2984
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997
This theorem is referenced by:  difsnss  3557  exmidundif  3999  undifdcss  6560
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