Theorem elpredgg 6273

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)

Assertion

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

Proof of Theorem elpredgg

Step	Hyp	Ref	Expression
1		df-pred 6260	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2	1	elin2 4144	. 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})))
3		elinisegg 6053	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
4	3	anbi2d 631	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ((𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
5	2, 4	bitrid 283	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  {csn 4568   class class class wbr 5086  ^◡ccnv 5624   “ cima 5628  Predcpred 6259

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 5232  ax-pr 5371

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 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260

This theorem is referenced by:  elpredg  6274  elpredimg  6275  elpred  6277