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Theorem undif1ss 3295
Description: Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undif1ss ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)

Proof of Theorem undif1ss
StepHypRef Expression
1 difss 3067 . 2 (𝐴𝐵) ⊆ 𝐴
2 unss1 3109 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵))
31, 2ax-mp 7 1 ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  cdif 2911  cun 2912  wss 2914
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 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928
This theorem is referenced by:  undif2ss  3296  pwundifss  4019
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