Theorem insucid 43645

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 3919	. 2 ⊢ (𝐴 ⊆ suc 𝐴 ↔ (𝐴 ∩ suc 𝐴) = 𝐴)
3	1, 2	mpbi 230	1 ⊢ (𝐴 ∩ suc 𝐴) = 𝐴

Syntax hints:   = wceq 1541   ∩ cin 3900   ⊆ wss 3901  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-in 3908  df-ss 3918  df-suc 6323

This theorem is referenced by: (None)