Theorem elpred 6321

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 6317	. 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ V) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
3	1, 2	mpan2 704	1 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102  Predcpred 6303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304

This theorem is used by:  predtrss  6325  setlikespec  6328  preddowncl  6335  xpord2pred  8165  xpord3pred  8172  fprlem2  8323  ttrclselem2  9731  ttrclse  9732  preduz  13788  predfz  13791  onsis  28666  ons2ind  28667  wzel  36679  wsuclem  36680