Theorem preex 39241

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 39224	. 2 ⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
2		iotaex 6509	. 2 ⊢ (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)) ∈ V
3	1, 2	eqeltri 2856	1 ⊢ pre 𝑁 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3450  dom cdm 5655  Predcpred 6298  ℩cio 6487   SucMap csucmap 38927   pre cpre 38929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6489  df-pre 39224

This theorem is used by:  presucmap  39244  preuniqval  39245  sucpre  39246  preel  39249