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Theorem undif2ss 3408
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 3407 . 2 ((𝐵𝐴) ∪ 𝐴) ⊆ (𝐵𝐴)
2 uncom 3190 . 2 (𝐴 ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ 𝐴)
3 uncom 3190 . 2 (𝐴𝐵) = (𝐵𝐴)
41, 2, 33sstr4i 3108 1 (𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  cdif 3038  cun 3039  wss 3041
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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