Theorem sucmapsuc 38601

Description: A set is succeeded by its successor. (Contributed by Peter Mazsa, 7-Jan-2026.)

Assertion

Ref	Expression
sucmapsuc	⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)

Proof of Theorem sucmapsuc

Step	Hyp	Ref	Expression
1		eqid 2734	. 2 ⊢ suc 𝑀 = suc 𝑀
2		sucexg 7748	. . 3 ⊢ (𝑀 ∈ 𝑉 → suc 𝑀 ∈ V)
3		brsucmap 38579	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ suc 𝑀 ∈ V) → (𝑀 SucMap suc 𝑀 ↔ suc 𝑀 = suc 𝑀))
4	2, 3	mpdan 687	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 SucMap suc 𝑀 ↔ suc 𝑀 = suc 𝑀))
5	1, 4	mpbiri 258	1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  suc csuc 6317   SucMap csucmap 38317

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-suc 6321  df-sucmap 38575

This theorem is referenced by:  presuc  38610