Theorem preex 38743

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 38726	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		iotaex 6476	. 2 ⊢ (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) ∈ V
3	1, 2	eqeltri 2833	1 ⊢ pre 𝑁 ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3442  dom cdm 5632  Predcpred 6266  ℩cio 6454   SucMap csucmap 38429   pre cpre 38431

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-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-sn 4583  df-pr 4585  df-uni 4866  df-iota 6456  df-pre 38726

This theorem is referenced by:  presucmap  38746  preuniqval  38747  sucpre  38748  preel  38751