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Theorem bnj1418 32437
 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 5034 . 2 (𝑧 = 𝑦 → (𝑧𝑅𝑥𝑦𝑅𝑥))
2 df-bnj14 32084 . . 3 pred(𝑥, 𝐴, 𝑅) = {𝑧𝐴𝑧𝑅𝑥}
32bnj1538 32252 . 2 (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
41, 3vtoclga 3522 1 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5031   predc-bnj14 32083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-bnj14 32084 This theorem is referenced by:  bnj1417  32438  bnj1523  32468
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