Theorem bnj1152 34843

Description: Technical lemma for bnj69 34855. 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) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))

Proof of Theorem bnj1152

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5156	. 2 ⊢ (𝑦 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑌𝑅𝑋))
2		df-bnj14 34534	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
3	1, 2	elrab2 3684	1 ⊢ (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2099   class class class wbr 5153   predc-bnj14 34533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-bnj14 34534

This theorem is referenced by:  bnj1175  34849  bnj1177  34851  bnj1388  34878