Theorem insucid 43855

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 6399	. 2 ⊢ 𝐴 ⊆ suc 𝐴
2		dfss2 3908	. 2 ⊢ (𝐴 ⊆ suc 𝐴 ↔ (𝐴 ∩ suc 𝐴) = 𝐴)
3	1, 2	mpbi 231	1 ⊢ (𝐴 ∩ suc 𝐴) = 𝐴

Syntax hints:   = wceq 1547   ∩ cin 3889   ⊆ wss 3890  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-in 3897  df-ss 3907  df-suc 6323

This theorem is referenced by: (None)