 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  undif2ss GIF version

Theorem undif2ss 3377
 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 3376 . 2 ((𝐵𝐴) ∪ 𝐴) ⊆ (𝐵𝐴)
2 uncom 3159 . 2 (𝐴 ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ 𝐴)
3 uncom 3159 . 2 (𝐴𝐵) = (𝐵𝐴)
41, 2, 33sstr4i 3080 1 (𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∖ cdif 3010   ∪ cun 3011   ⊆ wss 3013 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator