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Theorem elpredim 5609
 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 5597 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3763 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elpredim.1 . . . . 5 𝑋 ∈ V
4 elimasng 5410 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
5 opelcnvg 5224 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ 𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
64, 5bitrd 267 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
73, 6mpan 702 . . . 4 (𝑌 ∈ (𝑅 “ {𝑋}) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
87ibi 255 . . 3 (𝑌 ∈ (𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9 df-br 4584 . . 3 (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
108, 9sylibr 223 . 2 (𝑌 ∈ (𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
112, 10simplbiim 657 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173  {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:  predbrg  5617  preddowncl  5624  trpredrec  30982
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