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Theorem undifr 4483
Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.) (Proof shortened by SN, 11-Mar-2025.)
Assertion
Ref Expression
undifr (𝐴𝐵 ↔ ((𝐵𝐴) ∪ 𝐴) = 𝐵)

Proof of Theorem undifr
StepHypRef Expression
1 ssequn2 4189 . 2 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
2 undif1 4476 . . 3 ((𝐵𝐴) ∪ 𝐴) = (𝐵𝐴)
32eqeq1i 2742 . 2 (((𝐵𝐴) ∪ 𝐴) = 𝐵 ↔ (𝐵𝐴) = 𝐵)
41, 3bitr4i 278 1 (𝐴𝐵 ↔ ((𝐵𝐴) ∪ 𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  cdif 3948  cun 3949  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334
This theorem is referenced by:  difsnid  4810  f1ofvswap  7326  ralxpmap  8936  psdmullem  22169  psdmul  22170  tocyc01  33138  rprmdvdsprod  33562  aks6d1c5lem3  42138  selvvvval  42595  evlselvlem  42596  evlselv  42597  isubgr3stgrlem3  47935
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