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Theorem predexg 6215
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 6197 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 inex1g 5243 . 2 (𝐴𝑉 → (𝐴 ∩ (𝑅 “ {𝑋})) ∈ V)
31, 2eqeltrid 2843 1 (𝐴𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3431  cin 3887  {csn 4563  ccnv 5585  cima 5589  Predcpred 6196
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3433  df-in 3895  df-pred 6197
This theorem is referenced by:  predasetexOLD  6216
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