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Theorem elpredg 5692
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 5678 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3799 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
32baib 944 . . 3 (𝑌𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
43adantl 482 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
5 elimasng 5489 . . 3 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
6 df-br 4652 . . 3 (𝑋𝑅𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅)
75, 6syl6bbr 278 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑋𝑅𝑌))
8 brcnvg 5301 . 2 ((𝑋𝐵𝑌𝐴) → (𝑋𝑅𝑌𝑌𝑅𝑋))
94, 7, 83bitrd 294 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1989  {csn 4175  cop 4181   class class class wbr 4651  ccnv 5111  cima 5115  Predcpred 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678
This theorem is referenced by:  predpo  5696  predpoirr  5706  predfrirr  5707  wfrlem10  7421  wsuclem  31757  wsuclemOLD  31758  wsuclb  31761
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