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Theorem elpredg 5611
 Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
Assertion
Ref Expression
elpredg ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))

Proof of Theorem elpredg
StepHypRef Expression
1 df-pred 5597 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3763 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
32baib 942 . . 3 (𝑌𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
43adantl 481 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
5 elimasng 5410 . . 3 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
6 df-br 4584 . . 3 (𝑋𝑅𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅)
75, 6syl6bbr 277 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑋𝑅𝑌))
8 brcnvg 5225 . 2 ((𝑋𝐵𝑌𝐴) → (𝑋𝑅𝑌𝑌𝑅𝑋))
94, 7, 83bitrd 293 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  {csn 4125  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037   “ cima 5041  Predcpred 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597 This theorem is referenced by:  predpo  5615  predpoirr  5625  predfrirr  5626  wfrlem10  7311  wsuclem  31017  wsuclemOLD  31018  wsuclb  31021
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