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Theorem bnj1418 35054
Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1418 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Proof of Theorem bnj1418
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 breq1 5146 . 2 (𝑧 = 𝑦 → (𝑧𝑅𝑥𝑦𝑅𝑥))
2 df-bnj14 34703 . . 3 pred(𝑥, 𝐴, 𝑅) = {𝑧𝐴𝑧𝑅𝑥}
32bnj1538 34869 . 2 (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
41, 3vtoclga 3577 1 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108   class class class wbr 5143   predc-bnj14 34702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-bnj14 34703
This theorem is referenced by:  bnj1417  35055  bnj1523  35085
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