Theorem elpred 6337

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 6333	. 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 2107  Vcvv 3479   class class class wbr 5142  Predcpred 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320

This theorem is referenced by:  predtrss  6342  setlikespec  6345  preddowncl  6352  xpord2pred  8171  xpord3pred  8178  fprlem2  8327  wfrlem10OLD  8359  ttrclselem2  9767  ttrclse  9768  preduz  13691  predfz  13694  wzel  35826  wsuclem  35827