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Theorem bnj1152 32154
Description: Technical lemma for bnj69 32166. 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 5066 . 2 (𝑦 = 𝑌 → (𝑦𝑅𝑋𝑌𝑅𝑋))
2 df-bnj14 31845 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
31, 2elrab2 3687 1 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2107   class class class wbr 5063   predc-bnj14 31844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-bnj14 31845
This theorem is referenced by:  bnj1175  32160  bnj1177  32162  bnj1388  32189
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