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Theorem trpred 6325
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 6324 . . 3 (𝑋𝐴 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
21adantl 481 . 2 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
3 trss 5269 . . . 4 (Tr 𝐴 → (𝑋𝐴𝑋𝐴))
43imp 406 . . 3 ((Tr 𝐴𝑋𝐴) → 𝑋𝐴)
5 sseqin2 4210 . . 3 (𝑋𝐴 ↔ (𝐴𝑋) = 𝑋)
64, 5sylib 217 . 2 ((Tr 𝐴𝑋𝐴) → (𝐴𝑋) = 𝑋)
72, 6eqtrd 2766 1 ((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cin 3942  wss 3943  Tr wtr 5258   E cep 5572  Predcpred 6292
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293
This theorem is referenced by:  predon  7769  omsinds  7872
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