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Theorem predexg 6272
Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
Assertion
Ref Expression
predexg (𝐴𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Proof of Theorem predexg
StepHypRef Expression
1 df-pred 6254 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 inex1g 5277 . 2 (𝐴𝑉 → (𝐴 ∩ (𝑅 “ {𝑋})) ∈ V)
31, 2eqeltrid 2838 1 (𝐴𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444  cin 3910  {csn 4587  ccnv 5633  cima 5637  Predcpred 6253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-in 3918  df-pred 6254
This theorem is referenced by:  predasetexOLD  6273
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