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Theorem trpred 6337
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 6336 . . 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 4215 . . 3 (𝑋𝐴 ↔ (𝐴𝑋) = 𝑋)
64, 5sylib 217 . 2 ((Tr 𝐴𝑋𝐴) → (𝐴𝑋) = 𝑋)
72, 6eqtrd 2768 1 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cin 3946  wss 3947  Tr wtr 5265   E cep 5581  Predcpred 6304
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-10 2130  ax-11 2147  ax-12 2167  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-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  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-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5582  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305
This theorem is referenced by:  predon  7788  omsinds  7891
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