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Theorem elpredgg 6312
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 6299 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 4193 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elinisegg 6091 . . 3 ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
43anbi2d 628 . 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 395  wcel 2099  {csn 4624   class class class wbr 5142  ccnv 5671  cima 5675  Predcpred 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299
This theorem is referenced by:  elpredg  6313  elpredimg  6314  elpred  6316
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