Theorem insucid 42456

Description: The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)

Assertion

Ref	Expression
insucid	⊢ (𝐴 ∩ suc 𝐴) = 𝐴

Proof of Theorem insucid

Step	Hyp	Ref	Expression
1		sssucid 6444	. 2 ⊢ 𝐴 ⊆ suc 𝐴
2		df-ss 3965	. 2 ⊢ (𝐴 ⊆ suc 𝐴 ↔ (𝐴 ∩ suc 𝐴) = 𝐴)
3	1, 2	mpbi 229	1 ⊢ (𝐴 ∩ suc 𝐴) = 𝐴

Syntax hints:   = wceq 1541   ∩ cin 3947   ⊆ wss 3948  suc csuc 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-suc 6370

This theorem is referenced by: (None)