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Theorem bnj1152 31192
Description: Technical lemma for bnj69 31204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1152 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))

Proof of Theorem bnj1152
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq1 4688 . 2 (𝑦 = 𝑌 → (𝑦𝑅𝑋𝑌𝑅𝑋))
2 df-bnj14 30883 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
31, 2elrab2 3399 1 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030   class class class wbr 4685   predc-bnj14 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-bnj14 30883
This theorem is referenced by:  bnj1175  31198  bnj1177  31200  bnj1388  31227
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