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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 4354 | . 2 ⊢ suc ∅ = (∅ ∪ {∅}) | |
2 | uncom 3271 | . 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅) | |
3 | un0 3447 | . 2 ⊢ ({∅} ∪ ∅) = {∅} | |
4 | 1, 2, 3 | 3eqtri 2195 | 1 ⊢ suc ∅ = {∅} |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∪ cun 3119 ∅c0 3414 {csn 3581 suc csuc 4348 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-suc 4354 |
This theorem is referenced by: ordtriexmidlem 4501 ordtri2orexmid 4505 2ordpr 4506 onsucsssucexmid 4509 onsucelsucexmid 4512 ordsoexmid 4544 ordtri2or2exmid 4553 ontri2orexmidim 4554 nnregexmid 4603 omsinds 4604 tfr0dm 6298 df1o2 6405 nninfsellemdc 13965 |
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