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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sucdifsn | Structured version Visualization version GIF version |
Description: The difference between the successor and the singleton of a class is the class. (Contributed by Peter Mazsa, 20-Sep-2024.) |
Ref | Expression |
---|---|
sucdifsn | ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6364 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | difeq1i 4113 | . 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
3 | sucdifsn2 37617 | . 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴 | |
4 | 2, 3 | eqtri 2754 | 1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3940 ∪ cun 3941 {csn 4623 suc csuc 6360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-pr 5420 ax-reg 9589 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-nul 4318 df-sn 4624 df-pr 4626 df-suc 6364 |
This theorem is referenced by: partsuc 38163 |
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