Theorem eupre 38815

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 38790	. . 3 ⊢ Suc = ran SucMap
2	1	eleq2i 2828	. 2 ⊢ (𝑁 ∈ Suc ↔ 𝑁 ∈ ran SucMap )
3		eupre2 38814	. 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 2114  ∃!weu 2568   class class class wbr 5085  ran crn 5632   SucMap csucmap 38499   Suc csuccl 38500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689  ax-reg 9507

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-eprel 5531  df-fr 5584  df-cnv 5639  df-dm 5641  df-rn 5642  df-suc 6329  df-sucmap 38783  df-succl 38790

This theorem is referenced by: (None)