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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 4196 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elinisegg 6091 . . 3 ((𝑋𝑉𝑌𝑊) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
43anbi2d 627 . 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 2104  {csn 4627   class class class wbr 5147  ccnv 5674  cima 5678  Predcpred 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299
This theorem is referenced by:  elpredg  6313  elpredimg  6314  elpred  6316
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