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Theorem undif2ss 3513
Description: Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undif2ss (𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)

Proof of Theorem undif2ss
StepHypRef Expression
1 undif1ss 3512 . 2 ((𝐵𝐴) ∪ 𝐴) ⊆ (𝐵𝐴)
2 uncom 3294 . 2 (𝐴 ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ 𝐴)
3 uncom 3294 . 2 (𝐴𝐵) = (𝐵𝐴)
41, 2, 33sstr4i 3211 1 (𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  cdif 3141  cun 3142  wss 3144
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157
This theorem is referenced by: (None)
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