Theorem exeupre2 38781

Description: Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)

Assertion

Ref	Expression
exeupre2	⊢ (∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁)

Distinct variable group:   𝑚,𝑁

Proof of Theorem exeupre2

Step	Hyp	Ref	Expression
1		mopre 38780	. 2 ⊢ ∃*𝑚 suc 𝑚 = 𝑁
2		moeuex 2581	. 2 ⊢ (∃*𝑚 suc 𝑚 = 𝑁 → (∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁))
3	1, 2	ax-mp 5	1 ⊢ (∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁)

Syntax hints:   ↔ wb 206   = wceq 1542  ∃wex 1781  ∃*wmo 2536  ∃!weu 2567  suc csuc 6314

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 2707  ax-sep 5220  ax-pr 5364  ax-un 7678  ax-reg 9496

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-eprel 5520  df-fr 5573  df-suc 6318

This theorem is referenced by:  dfsuccl3  38782  exeupre  38800