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Theorem undif2ss 3293
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 3292 . 2 ((BA) ∪ A) ⊆ (BA)
2 uncom 3081 . 2 (A ∪ (BA)) = ((BA) ∪ A)
3 uncom 3081 . 2 (AB) = (BA)
41, 2, 33sstr4i 2978 1 (A ∪ (BA)) ⊆ (AB)
Colors of variables: wff set class
Syntax hints:  cdif 2908  cun 2909  wss 2911
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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