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Theorem bnj1418 31415
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 4807 . 2 (𝑧 = 𝑦 → (𝑧𝑅𝑥𝑦𝑅𝑥))
2 df-bnj14 31064 . . 3 pred(𝑥, 𝐴, 𝑅) = {𝑧𝐴𝑧𝑅𝑥}
32bnj1538 31232 . 2 (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
41, 3vtoclga 3412 1 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139   class class class wbr 4804   predc-bnj14 31063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-bnj14 31064
This theorem is referenced by:  bnj1417  31416  bnj1523  31446
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