Theorem dfpre3 38591

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 38590	. 2 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2		brsucmap 38579	. . . 4 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
3	2	el2v1 38364	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
4	3	iotabidv 6474	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 SucMap 𝑁) = (℩𝑚 suc 𝑚 = 𝑁))
5	1, 4	eqtrd 2769	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  suc csuc 6317  ℩cio 6444   SucMap csucmap 38317   pre cpre 38319

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 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-suc 6321  df-iota 6446  df-sucmap 38575  df-pre 38588

This theorem is referenced by: (None)