Theorem eupre 38993

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 38968	. . 3 ⊢ Suc = ran SucMap
2	1	eleq2i 2854	. 2 ⊢ (𝑁 ∈ Suc ↔ 𝑁 ∈ ran SucMap )
3		eupre2 38992	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
4	2, 3	bitrid 285	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2142  ∃!weu 2595   class class class wbr 5100  ran crn 5648   SucMap csucmap 38677   Suc csuccl 38678

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  ax-reg 9540

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-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  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-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-eprel 5547  df-fr 5600  df-cnv 5655  df-dm 5657  df-rn 5658  df-suc 6352  df-sucmap 38961  df-succl 38968

This theorem is referenced by: (None)