Theorem dfpre3 38501

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 38500	. 2 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))
2		brsucmap 38489	. . . 4 ⊢ ((𝑚 ∈ V ∧ 𝑁 ∈ 𝑉) → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
3	2	el2v1 38274	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑚 SucMap 𝑁 ↔ suc 𝑚 = 𝑁))
4	3	iotabidv 6465	. 2 ⊢ (𝑁 ∈ 𝑉 → (℩𝑚𝑚 SucMap 𝑁) = (℩𝑚 suc 𝑚 = 𝑁))
5	1, 4	eqtrd 2766	1 ⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  suc csuc 6308  ℩cio 6435   SucMap csucmap 38227   pre cpre 38229

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-suc 6312  df-iota 6437  df-sucmap 38485  df-pre 38498

This theorem is referenced by: (None)