Theorem preex 38665

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 38649	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		iotaex 6468	. 2 ⊢ (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) ∈ V
3	1, 2	eqeltri 2832	1 ⊢ pre 𝑁 ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3440  dom cdm 5624  Predcpred 6258  ℩cio 6446   SucMap csucmap 38378   pre cpre 38380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-sn 4581  df-pr 4583  df-uni 4864  df-iota 6448  df-pre 38649

This theorem is referenced by:  presucmap  38668  preuniqval  38669  sucpre  38670  preel  38673