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Theorem elpredim 5651
Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
Hypothesis
Ref Expression
elpredim.1 𝑋 ∈ V
Assertion
Ref Expression
elpredim (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Proof of Theorem elpredim
StepHypRef Expression
1 df-pred 5639 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3779 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elpredim.1 . . . . 5 𝑋 ∈ V
4 elimasng 5450 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
5 opelcnvg 5262 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ 𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
64, 5bitrd 268 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
73, 6mpan 705 . . . 4 (𝑌 ∈ (𝑅 “ {𝑋}) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
87ibi 256 . . 3 (𝑌 ∈ (𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9 df-br 4614 . . 3 (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
108, 9sylibr 224 . 2 (𝑌 ∈ (𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
112, 10simplbiim 658 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  Vcvv 3186  {csn 4148  cop 4154   class class class wbr 4613  ccnv 5073  cima 5077  Predcpred 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639
This theorem is referenced by:  predbrg  5659  preddowncl  5666  trpredrec  31436
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