Theorem preex 38830

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 38813	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		iotaex 6469	. 2 ⊢ (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) ∈ V
3	1, 2	eqeltri 2833	1 ⊢ pre 𝑁 ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3430  dom cdm 5625  Predcpred 6259  ℩cio 6447   SucMap csucmap 38516   pre cpre 38518

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 5242

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 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-sn 4569  df-pr 4571  df-uni 4852  df-iota 6449  df-pre 38813

This theorem is referenced by:  presucmap  38833  preuniqval  38834  sucpre  38835  preel  38838