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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ressucdifsn2 | Structured version Visualization version GIF version | ||
| Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38247. (Contributed by Peter Mazsa, 24-Jul-2024.) | 
| Ref | Expression | 
|---|---|
| ressucdifsn2 | ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | disjcsn 9645 | . 2 ⊢ (𝐴 ∩ {𝐴}) = ∅ | |
| 2 | disjresundif 38245 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 {csn 4625 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-reg 9633 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 df-rel 5691 df-res 5696 | 
| This theorem is referenced by: ressucdifsn 38247 partsuc2 38781 | 
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