Theorem mopre 38494

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 2753	. . . 4 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → suc 𝑚 = suc 𝑙)
2		suc11reg 9509	. . . 4 ⊢ (suc 𝑚 = suc 𝑙 ↔ 𝑚 = 𝑙)
3	1, 2	sylib 218	. . 3 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
4	3	gen2 1797	. 2 ⊢ ∀𝑚∀𝑙((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
5		suceq 6374	. . . 4 ⊢ (𝑚 = 𝑙 → suc 𝑚 = suc 𝑙)
6	5	eqeq1d 2733	. . 3 ⊢ (𝑚 = 𝑙 → (suc 𝑚 = 𝑁 ↔ suc 𝑙 = 𝑁))
7	6	mo4 2561	. 2 ⊢ (∃*𝑚 suc 𝑚 = 𝑁 ↔ ∀𝑚∀𝑙((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙))
8	4, 7	mpbir 231	1 ⊢ ∃*𝑚 suc 𝑚 = 𝑁

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃*wmo 2533  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-eprel 5514  df-fr 5567  df-suc 6312

This theorem is referenced by:  exeupre2  38495