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Theorem trpred 6354
Description: The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)
Assertion
Ref Expression
trpred ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

Proof of Theorem trpred
StepHypRef Expression
1 predep 6353 . . 3 (𝑋𝐴 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
21adantl 481 . 2 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
3 trss 5276 . . . 4 (Tr 𝐴 → (𝑋𝐴𝑋𝐴))
43imp 406 . . 3 ((Tr 𝐴𝑋𝐴) → 𝑋𝐴)
5 sseqin2 4231 . . 3 (𝑋𝐴 ↔ (𝐴𝑋) = 𝑋)
64, 5sylib 218 . 2 ((Tr 𝐴𝑋𝐴) → (𝐴𝑋) = 𝑋)
72, 6eqtrd 2775 1 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963  Tr wtr 5265   E cep 5588  Predcpred 6322
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  predon  7805  omsinds  7908
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