Theorem eupre 39426

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 39401	. . 3 ⊢ Suc = ran SucMap
2	1	eleq2i 2853	. 2 ⊢ (𝑁 ∈ Suc ↔ 𝑁 ∈ ran SucMap )
3		eupre2 39425	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))
4	2, 3	bitrid 286	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  ∃!weu 2594   class class class wbr 5103  ran crn 5652   SucMap csucmap 39110   Suc csuccl 39111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7751  ax-reg 9586

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-eprel 5551  df-fr 5604  df-cnv 5659  df-dm 5661  df-rn 5662  df-suc 6368  df-sucmap 39394  df-succl 39401

This theorem is used by: (None)