Theorem sucunisn 43353

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 4891	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		suceq 6401	. 2 ⊢ (∪ {𝐴} = 𝐴 → suc ∪ {𝐴} = suc 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4591  ∪ cuni 4873  suc csuc 6336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3921  df-ss 3933  df-sn 4592  df-pr 4594  df-uni 4874  df-suc 6340

This theorem is referenced by: (None)