Theorem sucunisn 44158

Description: The successor to the union of any singleton of a set is the successor of the set. (Contributed by RP, 11-Feb-2025.)

Assertion

Ref	Expression
sucunisn	⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)

Proof of Theorem sucunisn

Step	Hyp	Ref	Expression
1		unisng 4892	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		suceq 6433	. 2 ⊢ (∪ {𝐴} = 𝐴 → suc ∪ {𝐴} = suc 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {csn 4591  ∪ cuni 4874  suc csuc 6366

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-sn 4592  df-pr 4594  df-uni 4875  df-suc 6370

This theorem is used by: (None)