Theorem elpred 6340

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.)

Hypothesis

Ref	Expression
elpred.1	⊢ 𝑌 ∈ V

Assertion

Ref	Expression
elpred	⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

Proof of Theorem elpred

Step	Hyp	Ref	Expression
1		elpred.1	. 2 ⊢ 𝑌 ∈ V
2		elpredgg 6336	. 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ V) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
3	1, 2	mpan2 691	1 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148  Predcpred 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323

This theorem is referenced by:  predtrss  6345  setlikespec  6348  preddowncl  6355  xpord2pred  8169  xpord3pred  8176  fprlem2  8325  wfrlem10OLD  8357  ttrclselem2  9764  ttrclse  9765  preduz  13687  predfz  13690  wzel  35806  wsuclem  35807