Theorem exeupre2 38976

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 38975	. 2 ⊢ ∃*𝑚 suc 𝑚 = 𝑁
2		moeuex 2611	. 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 208   = wceq 1562  ∃wex 1801  ∃*wmo 2566  ∃!weu 2597  suc csuc 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720  ax-reg 9542

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-eprel 5549  df-fr 5602  df-suc 6354

This theorem is referenced by:  dfsuccl3  38977  exeupre  38995