Theorem dfpre3 39016

Description: Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)

Assertion

Ref	Expression
dfpre3	⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))

Distinct variable groups:   𝑚,𝑁   𝑚,𝑉

Proof of Theorem dfpre3

Step	Hyp	Ref	Expression
1		dfpre2 39015	. 2 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2		brsucmap 39004	. . . 4 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
3	2	el2v1 38767	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
4	3	iotabidv 6521	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 SucMap 𝑁) = (℩𝑚 suc 𝑚 = 𝑁))
5	1, 4	eqtrd 2804	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  Vcvv 3463   class class class wbr 5113  suc csuc 6363  ℩cio 6491   SucMap csucmap 38716   pre cpre 38718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-suc 6367  df-iota 6493  df-sucmap 39000  df-pre 39013

This theorem is referenced by: (None)