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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 6370 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | difeq1i 4118 | . 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
3 | sucdifsn2 37099 | . 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴 | |
4 | 2, 3 | eqtri 2760 | 1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∖ cdif 3945 ∪ cun 3946 {csn 4628 suc csuc 6366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-pr 5427 ax-reg 9586 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-nul 4323 df-sn 4629 df-pr 4631 df-suc 6370 |
This theorem is referenced by: partsuc 37645 |
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