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Theorem elpredgg 6323
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)
Assertion
Ref Expression
elpredgg ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))

Proof of Theorem elpredgg
StepHypRef Expression
1 df-pred 6310 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 4199 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elinisegg 6102 . . 3 ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
43anbi2d 628 . 2 ((𝑋𝑉𝑌𝑊) → ((𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})) ↔ (𝑌𝐴𝑌𝑅𝑋)))
52, 4bitrid 282 1 ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  {csn 4632   class class class wbr 5152  ccnv 5681  cima 5685  Predcpred 6309
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310
This theorem is referenced by:  elpredg  6324  elpredimg  6325  elpred  6327
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