Theorem mopre 39180

Description: There is at most one predecessor of 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)

Assertion

Ref	Expression
mopre	⊢ ∃*𝑚 suc 𝑚 = 𝑁

Distinct variable group:   𝑚,𝑁

Proof of Theorem mopre

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2787	. . . 4 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → suc 𝑚 = suc 𝑙)
2		suc11reg 9595	. . . 4 ⊢ (suc 𝑚 = suc 𝑙 ↔ 𝑚 = 𝑙)
3	1, 2	sylib 221	. . 3 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
4	3	gen2 1829	. 2 ⊢ ∀𝑚∀𝑙((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
5		suceq 6433	. . . 4 ⊢ (𝑚 = 𝑙 → suc 𝑚 = suc 𝑙)
6	5	eqeq1d 2767	. . 3 ⊢ (𝑚 = 𝑙 → (suc 𝑚 = 𝑁 ↔ suc 𝑙 = 𝑁))
7	6	mo4 2596	. 2 ⊢ (∃*𝑚 suc 𝑚 = 𝑁 ↔ ∀𝑚∀𝑙((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙))
8	4, 7	mpbir 234	1 ⊢ ∃*𝑚 suc 𝑚 = 𝑁

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃*wmo 2567  suc csuc 6366

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742  ax-reg 9561

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 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-eprel 5563  df-fr 5616  df-suc 6370

This theorem is used by:  exeupre2  39181