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Theorem predbrg 6161
Description: Closed form of elpredim 6153. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
Assertion
Ref Expression
predbrg ((𝑋𝑉𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Proof of Theorem predbrg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 predeq3 6145 . . . . 5 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
21eleq2d 2895 . . . 4 (𝑥 = 𝑋 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
3 breq2 5061 . . . 4 (𝑥 = 𝑋 → (𝑌𝑅𝑥𝑌𝑅𝑋))
42, 3imbi12d 346 . . 3 (𝑥 = 𝑋 → ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) ↔ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)))
5 vex 3495 . . . 4 𝑥 ∈ V
65elpredim 6153 . . 3 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥)
74, 6vtoclg 3565 . 2 (𝑋𝑉 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋))
87imp 407 1 ((𝑋𝑉𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   class class class wbr 5057  Predcpred 6140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141
This theorem is referenced by: (None)
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