Theorem sucmapsuc 39238

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 2760	. 2 ⊢ suc 𝑀 = suc 𝑀
2		sucexg 7805	. . 3 ⊢ (𝑀 ∈ 𝑉 → suc 𝑀 ∈ V)
3		brsucmap 39215	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ suc 𝑀 ∈ V) → (𝑀 SucMap suc 𝑀 ↔ suc 𝑀 = suc 𝑀))
4	2, 3	mpdan 700	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 SucMap suc 𝑀 ↔ suc 𝑀 = suc 𝑀))
5	1, 4	mpbiri 261	1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3450   class class class wbr 5103  suc csuc 6359   SucMap csucmap 38927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-suc 6363  df-sucmap 39211

This theorem is used by:  presuc  39247