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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 6190 | . 2 ⊢ suc ∅ = (∅ ∪ {∅}) | |
2 | uncom 4126 | . 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅) | |
3 | un0 4341 | . 2 ⊢ ({∅} ∪ ∅) = {∅} | |
4 | 1, 2, 3 | 3eqtri 2845 | 1 ⊢ suc ∅ = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∪ cun 3931 ∅c0 4288 {csn 4557 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 df-un 3938 df-nul 4289 df-suc 6190 |
This theorem is referenced by: df1o2 8105 axdc3lem4 9863 |
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