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Theorem predbrg 6166
Description: Closed form of elpredim 6158. (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 6150 . . . . 5 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
21eleq2d 2903 . . . 4 (𝑥 = 𝑋 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
3 breq2 5067 . . . 4 (𝑥 = 𝑋 → (𝑌𝑅𝑥𝑌𝑅𝑋))
42, 3imbi12d 346 . . 3 (𝑥 = 𝑋 → ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) ↔ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)))
5 vex 3503 . . . 4 𝑥 ∈ V
65elpredim 6158 . . 3 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥)
74, 6vtoclg 3573 . 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 1530  wcel 2107   class class class wbr 5063  Predcpred 6145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146
This theorem is referenced by: (None)
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