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Theorem predexg 6311
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 6293 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 inex1g 5312 . 2 (𝐴𝑉 → (𝐴 ∩ (𝑅 “ {𝑋})) ∈ V)
31, 2eqeltrid 2831 1 (𝐴𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3468  cin 3942  {csn 4623  ccnv 5668  cima 5672  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-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-pred 6293
This theorem is referenced by:  predasetexOLD  6312
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