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Theorem ecref 32427
Description: All elements are in their own equivalence class. (Contributed by Thierry Arnoux, 14-Feb-2025.)
Assertion
Ref Expression
ecref ((𝑅 Er 𝑋𝐴𝑋) → 𝐴 ∈ [𝐴]𝑅)

Proof of Theorem ecref
StepHypRef Expression
1 simpl 482 . . 3 ((𝑅 Er 𝑋𝐴𝑋) → 𝑅 Er 𝑋)
2 simpr 484 . . 3 ((𝑅 Er 𝑋𝐴𝑋) → 𝐴𝑋)
31, 2erref 8720 . 2 ((𝑅 Er 𝑋𝐴𝑋) → 𝐴𝑅𝐴)
4 elecg 8743 . . 3 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
52, 4sylancom 587 . 2 ((𝑅 Er 𝑋𝐴𝑋) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
63, 5mpbird 257 1 ((𝑅 Er 𝑋𝐴𝑋) → 𝐴 ∈ [𝐴]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098   class class class wbr 5139   Er wer 8697  [cec 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-er 8700  df-ec 8702
This theorem is referenced by:  ghmquskerlem1  33023  ghmquskerlem2  33025  qsdrnglem2  33105
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