Theorem mopre 38809

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 2759	. . . 4 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → suc 𝑚 = suc 𝑙)
2		suc11reg 9534	. . . 4 ⊢ (suc 𝑚 = suc 𝑙 ↔ 𝑚 = 𝑙)
3	1, 2	sylib 218	. . 3 ⊢ ((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
4	3	gen2 1798	. 2 ⊢ ∀𝑚∀𝑙((suc 𝑚 = 𝑁 ∧ suc 𝑙 = 𝑁) → 𝑚 = 𝑙)
5		suceq 6386	. . . 4 ⊢ (𝑚 = 𝑙 → suc 𝑚 = suc 𝑙)
6	5	eqeq1d 2739	. . 3 ⊢ (𝑚 = 𝑙 → (suc 𝑚 = 𝑁 ↔ suc 𝑙 = 𝑁))
7	6	mo4 2567	. 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 1540   = wceq 1542  ∃*wmo 2538  suc csuc 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371  ax-un 7683  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-eprel 5525  df-fr 5578  df-suc 6324

This theorem is referenced by:  exeupre2  38810