Theorem elpredim 6319

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) (Proof shortened by BJ, 16-Oct-2024.)

Hypothesis

Ref	Expression
elpredim.1	⊢ 𝑋 ∈ V

Assertion

Ref	Expression
elpredim	⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Proof of Theorem elpredim

Step	Hyp	Ref	Expression
1		elpredim.1	. 2 ⊢ 𝑋 ∈ V
2		elpredimg 6318	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
3	1, 2	mpan 703	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453   class class class wbr 5107  Predcpred 6302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303

This theorem is used by:  preddowncl  6334