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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 6392 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | difeq1i 4132 | . 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
3 | sucdifsn2 38219 | . 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴 | |
4 | 2, 3 | eqtri 2763 | 1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∪ cun 3961 {csn 4631 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-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-nul 4340 df-sn 4632 df-pr 4634 df-suc 6392 |
This theorem is referenced by: partsuc 38762 |
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