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Mirrors > Home > MPE Home > Th. List > Mathboxes > insucid | Structured version Visualization version GIF version |
Description: The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.) |
Ref | Expression |
---|---|
insucid | ⊢ (𝐴 ∩ suc 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssucid 6466 | . 2 ⊢ 𝐴 ⊆ suc 𝐴 | |
2 | dfss2 3981 | . 2 ⊢ (𝐴 ⊆ suc 𝐴 ↔ (𝐴 ∩ suc 𝐴) = 𝐴) | |
3 | 1, 2 | mpbi 230 | 1 ⊢ (𝐴 ∩ suc 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3962 ⊆ wss 3963 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-suc 6392 |
This theorem is referenced by: (None) |
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