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Theorem undif2ss 3299
 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 3298 . 2 ((𝐵𝐴) ∪ 𝐴) ⊆ (𝐵𝐴)
2 uncom 3087 . 2 (𝐴 ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ 𝐴)
3 uncom 3087 . 2 (𝐴𝐵) = (𝐵𝐴)
41, 2, 33sstr4i 2984 1 (𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∖ cdif 2914   ∪ cun 2915   ⊆ wss 2917 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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