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Theorem undif1ss 3488
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 3253 . 2 (𝐴𝐵) ⊆ 𝐴
2 unss1 3296 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵))
31, 2ax-mp 5 1 ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  cdif 3118  cun 3119  wss 3121
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134
This theorem is referenced by:  undif2ss  3489  pwundifss  4268
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