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Theorem elpredg 5663
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 5649 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3785 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
32baib 943 . . 3 (𝑌𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
43adantl 482 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
5 elimasng 5460 . . 3 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
6 df-br 4624 . . 3 (𝑋𝑅𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅)
75, 6syl6bbr 278 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑋𝑅𝑌))
8 brcnvg 5273 . 2 ((𝑋𝐵𝑌𝐴) → (𝑋𝑅𝑌𝑌𝑅𝑋))
94, 7, 83bitrd 294 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  {csn 4155  cop 4161   class class class wbr 4623  ccnv 5083  cima 5087  Predcpred 5648
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 4751  ax-nul 4759  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649
This theorem is referenced by:  predpo  5667  predpoirr  5677  predfrirr  5678  wfrlem10  7384  wsuclem  31527  wsuclemOLD  31528  wsuclb  31531
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