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| Mirrors > Home > MPE Home > Th. List > suc0 | Structured version Visualization version GIF version | ||
| Description: The successor of the empty set. (Contributed by NM, 1-Feb-2005.) | 
| Ref | Expression | 
|---|---|
| suc0 | ⊢ suc ∅ = {∅} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-suc 6390 | . 2 ⊢ suc ∅ = (∅ ∪ {∅}) | |
| 2 | uncom 4158 | . 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅) | |
| 3 | un0 4394 | . 2 ⊢ ({∅} ∪ ∅) = {∅} | |
| 4 | 1, 2, 3 | 3eqtri 2769 | 1 ⊢ suc ∅ = {∅} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∪ cun 3949 ∅c0 4333 {csn 4626 suc csuc 6386 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-suc 6390 | 
| This theorem is referenced by: df1o2 8513 axdc3lem4 10493 pw2bday 28418 | 
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