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Theorem undif2ss 3276
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 3275 . 2 ((BA) ∪ A) ⊆ (BA)
2 uncom 3064 . 2 (A ∪ (BA)) = ((BA) ∪ A)
3 uncom 3064 . 2 (AB) = (BA)
41, 2, 33sstr4i 2961 1 (A ∪ (BA)) ⊆ (AB)
Colors of variables: wff set class
Syntax hints:  cdif 2891  cun 2892  wss 2894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908
This theorem is referenced by: (None)
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