Theorem insucid 44389

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		dfss2 3917	. 2 ⊢ (𝐴 ⊆ suc 𝐴 ↔ (𝐴 ∩ suc 𝐴) = 𝐴)
3	1, 2	mpbi 233	1 ⊢ (𝐴 ∩ suc 𝐴) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3898   ⊆ wss 3899  suc csuc 6363

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-in 3906  df-ss 3916  df-suc 6367

This theorem is used by: (None)