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Theorem undif2ss 3442
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 3441 . 2 ((𝐵𝐴) ∪ 𝐴) ⊆ (𝐵𝐴)
2 uncom 3224 . 2 (𝐴 ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ 𝐴)
3 uncom 3224 . 2 (𝐴𝐵) = (𝐵𝐴)
41, 2, 33sstr4i 3142 1 (𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  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: (None)
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