Theorem sucunisn 43555

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 4879	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		suceq 6383	. 2 ⊢ (∪ {𝐴} = 𝐴 → suc ∪ {𝐴} = suc 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {csn 4578  ∪ cuni 4861  suc csuc 6317

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-un 3904  df-ss 3916  df-sn 4579  df-pr 4581  df-uni 4862  df-suc 6321

This theorem is referenced by: (None)