Theorem sucmapsuc 38740

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 2737	. 2 ⊢ suc 𝑀 = suc 𝑀
2		sucexg 7760	. . 3 ⊢ (𝑀 ∈ 𝑉 → suc 𝑀 ∈ V)
3		brsucmap 38717	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ suc 𝑀 ∈ V) → (𝑀 SucMap suc 𝑀 ↔ suc 𝑀 = suc 𝑀))
4	2, 3	mpdan 688	. 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 1542   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100  suc csuc 6327   SucMap csucmap 38429

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 5243  ax-pr 5379  ax-un 7690

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-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-suc 6331  df-sucmap 38713

This theorem is referenced by:  presuc  38749