Theorem elpredimg 6274

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.) (Proof shortened by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
elpredimg	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Proof of Theorem elpredimg

Step	Hyp	Ref	Expression
1		elpredgg 6272	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
2		simpr 484	. . 3 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋) → 𝑌𝑅𝑋)
3	1, 2	biimtrdi 253	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋))
4	3	syldbl2 842	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   class class class wbr 5086  Predcpred 6258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259

This theorem is referenced by:  elpredim  6275  predtrss  6280