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Theorem elpredgg 6267
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 6254 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 4158 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elinisegg 6046 . . 3 ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
43anbi2d 630 . 2 ((𝑋𝑉𝑌𝑊) → ((𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})) ↔ (𝑌𝐴𝑌𝑅𝑋)))
52, 4bitrid 283 1 ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  {csn 4587   class class class wbr 5106  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  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254
This theorem is referenced by:  elpredg  6268  elpredimg  6269  elpred  6271
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