Theorem eupre 38516

Description: Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)

Assertion

Ref	Expression
eupre	⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))

Distinct variable groups:   𝑚,𝑁   𝑚,𝑉

Proof of Theorem eupre

Step	Hyp	Ref	Expression
1		df-succl 38492	. . 3 ⊢ Suc = ran SucMap
2	1	eleq2i 2823	. 2 ⊢ (𝑁 ∈ Suc ↔ 𝑁 ∈ ran SucMap )
3		eupre2 38515	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
4	2, 3	bitrid 283	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2111  ∃!weu 2563   class class class wbr 5089  ran crn 5615   SucMap csucmap 38227   Suc csuccl 38228

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  ax-reg 9478

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-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  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-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-eprel 5514  df-fr 5567  df-cnv 5622  df-dm 5624  df-rn 5625  df-suc 6312  df-sucmap 38485  df-succl 38492

This theorem is referenced by: (None)