Theorem mopre 39140

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 2785	. . . 4 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → suc 𝑚 = suc 𝑙)
2		suc11reg 9584	. . . 4 ⊢ (suc 𝑚 = suc 𝑙 ↔ 𝑚 = 𝑙)
3	1, 2	sylib 221	. . 3 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
4	3	gen2 1826	. 2 ⊢ ∀𝑚∀𝑙((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
5		suceq 6429	. . . 4 ⊢ (𝑚 = 𝑙 → suc 𝑚 = suc 𝑙)
6	5	eqeq1d 2765	. . 3 ⊢ (𝑚 = 𝑙 → (suc 𝑚 = 𝑁 ↔ suc 𝑙 = 𝑁))
7	6	mo4 2594	. 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

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃*wmo 2565  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-eprel 5561  df-fr 5614  df-suc 6366

This theorem is referenced by:  exeupre2  39141