Theorem exeupre2 39067

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 39066	. 2 ⊢ ∃*𝑚 suc 𝑚 = 𝑁
2		moeuex 2608	. 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 209   = wceq 1568  ∃wex 1807  ∃*wmo 2563  ∃!weu 2594  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404  ax-un 7732  ax-reg 9553

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  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 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-eprel 5561  df-fr 5614  df-suc 6366

This theorem is referenced by:  dfsuccl3  39068  exeupre  39086