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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 4389 | . 2 ⊢ suc ∅ = (∅ ∪ {∅}) | |
2 | uncom 3294 | . 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅) | |
3 | un0 3471 | . 2 ⊢ ({∅} ∪ ∅) = {∅} | |
4 | 1, 2, 3 | 3eqtri 2214 | 1 ⊢ suc ∅ = {∅} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∪ cun 3142 ∅c0 3437 {csn 3607 suc csuc 4383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-nul 3438 df-suc 4389 |
This theorem is referenced by: ordtriexmidlem 4536 ordtri2orexmid 4540 2ordpr 4541 onsucsssucexmid 4544 onsucelsucexmid 4547 ordsoexmid 4579 ordtri2or2exmid 4588 ontri2orexmidim 4589 nnregexmid 4638 omsinds 4639 tfr0dm 6346 df1o2 6453 nninfsellemdc 15213 |
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