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Theorem undifr 4489
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 4199 . 2 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
2 undif1 4482 . . 3 ((𝐵𝐴) ∪ 𝐴) = (𝐵𝐴)
32eqeq1i 2740 . 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 1537  cdif 3960  cun 3961  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340
This theorem is referenced by:  difsnid  4815  f1ofvswap  7326  ralxpmap  8935  psdmullem  22187  psdmul  22188  tocyc01  33121  rprmdvdsprod  33542  aks6d1c5lem3  42119  selvvvval  42572  evlselvlem  42573  evlselv  42574  isubgr3stgrlem3  47871
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