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Theorem elpredim 5877
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 5865 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3963 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elpredim.1 . . . . 5 𝑋 ∈ V
4 elimasng 5673 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
5 opelcnvg 5470 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ 𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
64, 5bitrd 270 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
73, 6mpan 681 . . . 4 (𝑌 ∈ (𝑅 “ {𝑋}) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
87ibi 258 . . 3 (𝑌 ∈ (𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9 df-br 4810 . . 3 (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
108, 9sylibr 225 . 2 (𝑌 ∈ (𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
112, 10simplbiim 499 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  Vcvv 3350  {csn 4334  cop 4340   class class class wbr 4809  ccnv 5276  cima 5280  Predcpred 5864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865
This theorem is referenced by:  predbrg  5885  preddowncl  5892  trpredrec  32181
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