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Theorem elpredim 5730
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 5718 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3834 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elpredim.1 . . . . 5 𝑋 ∈ V
4 elimasng 5526 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ (𝑅 “ {𝑋})) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
5 opelcnvg 5334 . . . . . 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 706 . . . 4 (𝑌 ∈ (𝑅 “ {𝑋}) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
87ibi 256 . . 3 (𝑌 ∈ (𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9 df-br 4686 . . 3 (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
108, 9sylibr 224 . 2 (𝑌 ∈ (𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
112, 10simplbiim 659 1 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  Vcvv 3231  {csn 4210  cop 4216   class class class wbr 4685  ccnv 5142  cima 5146  Predcpred 5717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718
This theorem is referenced by:  predbrg  5738  preddowncl  5745  trpredrec  31862
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