Theorem preex 39561

Description: The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)

Assertion

Ref	Expression
preex	⊢ pre 𝑁 ∈ V

Proof of Theorem preex

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pre 39544	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		iotaex 6518	. 2 ⊢ (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) ∈ V
3	1, 2	eqeltri 2856	1 ⊢ pre 𝑁 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450  dom cdm 5651  Predcpred 6306  ℩cio 6495   SucMap csucmap 39247   pre cpre 39249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6497  df-pre 39544

This theorem is used by:  presucmap  39564  preuniqval  39565  sucpre  39566  preel  39569