Theorem predexg 6317

Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)

Assertion

Ref	Expression
predexg	⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Proof of Theorem predexg

Step	Hyp	Ref	Expression
1		df-pred 6299	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inex1g 5287	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V)
3	1, 2	eqeltrid 2873	1 ⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  {csn 4591  ^◡ccnv 5658   “ cima 5662  Predcpred 6298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920  df-pred 6299

This theorem is referenced by: (None)