Theorem exeupre2 39384

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 39383	. 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)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812  ∃*wmo 2563  ∃!weu 2594  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7749  ax-reg 9579

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  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 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-eprel 5551  df-fr 5604  df-suc 6367

This theorem is used by:  dfsuccl3  39385  exeupre  39403