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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 6359 | . 2 ⊢ suc ∅ = (∅ ∪ {∅}) | |
2 | uncom 4149 | . 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅) | |
3 | un0 4386 | . 2 ⊢ ({∅} ∪ ∅) = {∅} | |
4 | 1, 2, 3 | 3eqtri 2763 | 1 ⊢ suc ∅ = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∪ cun 3942 ∅c0 4318 {csn 4622 suc csuc 6355 |
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-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-dif 3947 df-un 3949 df-nul 4319 df-suc 6359 |
This theorem is referenced by: df1o2 8455 axdc3lem4 10430 |
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