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Mirrors > Home > ILE Home > Th. List > suc0 | 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 4288 | . 2 ⊢ suc ∅ = (∅ ∪ {∅}) | |
2 | uncom 3215 | . 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅) | |
3 | un0 3391 | . 2 ⊢ ({∅} ∪ ∅) = {∅} | |
4 | 1, 2, 3 | 3eqtri 2162 | 1 ⊢ suc ∅ = {∅} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∪ cun 3064 ∅c0 3358 {csn 3522 suc csuc 4282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-un 3070 df-nul 3359 df-suc 4288 |
This theorem is referenced by: ordtriexmidlem 4430 ordtri2orexmid 4433 2ordpr 4434 onsucsssucexmid 4437 onsucelsucexmid 4440 ordsoexmid 4472 ordtri2or2exmid 4481 nnregexmid 4529 omsinds 4530 tfr0dm 6212 df1o2 6319 nninfsellemdc 13195 |
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