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Theorem elpredg 5879
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 5865 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3963 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
32baib 531 . . 3 (𝑌𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
43adantl 473 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
5 elimasng 5673 . . 3 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
6 df-br 4810 . . 3 (𝑋𝑅𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅)
75, 6syl6bbr 280 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑋𝑅𝑌))
8 brcnvg 5471 . 2 ((𝑋𝐵𝑌𝐴) → (𝑋𝑅𝑌𝑌𝑅𝑋))
94, 7, 83bitrd 296 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  {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 2069  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 2062  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:  predpo  5883  predpoirr  5893  predfrirr  5894  wfrlem10  7628  wsuclem  32146  wsuclb  32149
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