Theorem exeupre2 39513

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 39512	. 2 ⊢ ∃*𝑚 suc 𝑚 = 𝑁
2		moeuex 2607	. 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)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1811  ∃*wmo 2562  ∃!weu 2593  suc csuc 6367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391  ax-un 7755  ax-reg 9593

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-eprel 5551  df-fr 5604  df-suc 6371

This theorem is used by:  dfsuccl3  39514  exeupre  39532