Theorem elpred 6269

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 6265	. 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ V) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
3	1, 2	mpan2 697	1 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3431   class class class wbr 5072  Predcpred 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252

This theorem is referenced by:  predtrss  6273  setlikespec  6276  preddowncl  6283  xpord2pred  8085  xpord3pred  8092  fprlem2  8241  ttrclselem2  9638  ttrclse  9639  preduz  13595  predfz  13598  onsis  28284  ons2ind  28285  wzel  36050  wsuclem  36051