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Theorem bnj1418 31922
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 4971 . 2 (𝑧 = 𝑦 → (𝑧𝑅𝑥𝑦𝑅𝑥))
2 df-bnj14 31572 . . 3 pred(𝑥, 𝐴, 𝑅) = {𝑧𝐴𝑧𝑅𝑥}
32bnj1538 31739 . 2 (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
41, 3vtoclga 3519 1 (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2083   class class class wbr 4968   predc-bnj14 31571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-bnj14 31572
This theorem is referenced by:  bnj1417  31923  bnj1523  31953
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