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Theorem ecref 8768
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 8744 . 2 ((𝑅 Er 𝑋𝐴𝑋) → 𝐴𝑅𝐴)
4 elecg 8767 . . 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 2099   class class class wbr 5148   Er wer 8721  [cec 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-er 8724  df-ec 8726
This theorem is referenced by:  ghmquskerlem1  19233  ghmquskerlem2  19235  ghmqusnsglem1  33129  ghmqusnsglem2  33130  qsdrnglem2  33207
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