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Theorem undifss 3503
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 3261 . . . 4 (𝐵𝐴) ⊆ 𝐵
21jctr 315 . . 3 (𝐴𝐵 → (𝐴𝐵 ∧ (𝐵𝐴) ⊆ 𝐵))
3 unss 3309 . . 3 ((𝐴𝐵 ∧ (𝐵𝐴) ⊆ 𝐵) ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
42, 3sylib 122 . 2 (𝐴𝐵 → (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
5 ssun1 3298 . . 3 𝐴 ⊆ (𝐴 ∪ (𝐵𝐴))
6 sstr 3163 . . 3 ((𝐴 ⊆ (𝐴 ∪ (𝐵𝐴)) ∧ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵) → 𝐴𝐵)
75, 6mpan 424 . 2 ((𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵𝐴𝐵)
84, 7impbii 126 1 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  cdif 3126  cun 3127  wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142
This theorem is referenced by:  difsnss  3737  exmidundif  4203  exmidundifim  4204  undifdcss  6915
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