Theorem dfpre3 38977

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 38976	. 2 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2		brsucmap 38965	. . . 4 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
3	2	el2v1 38728	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
4	3	iotabidv 6505	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 SucMap 𝑁) = (℩𝑚 suc 𝑚 = 𝑁))
5	1, 4	eqtrd 2797	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100  suc csuc 6348  ℩cio 6475   SucMap csucmap 38677   pre cpre 38679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-suc 6352  df-iota 6477  df-sucmap 38961  df-pre 38974

This theorem is referenced by: (None)