Theorem elpredg 6307

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
elpredg	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))

Proof of Theorem elpredg

Step	Hyp	Ref	Expression
1		elpredgg 6306	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
2		ibar 528	. . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝑌𝑅𝑋 ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
3	2	bicomd 222	. . 3 ⊢ (𝑌 ∈ 𝐴 → ((𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
4	3	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → ((𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋))
5	1, 4	bitrd 279	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098   class class class wbr 5141  Predcpred 6292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293

This theorem is referenced by:  predpoirr  6327  predfrirr  6328  wfrlem10OLD  8316  wsuclem  35330  wsuclb  35333