Theorem sucunisn 43346

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 4905	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		suceq 6430	. 2 ⊢ (∪ {𝐴} = 𝐴 → suc ∪ {𝐴} = suc 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {csn 4606  ∪ cuni 4887  suc csuc 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-un 3936  df-ss 3948  df-sn 4607  df-pr 4609  df-uni 4888  df-suc 6369

This theorem is referenced by: (None)