![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > uni0 | GIF version |
Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.) |
Ref | Expression |
---|---|
uni0 | ⊢ ∪ ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3325 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | uni0b 3684 | . 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅}) | |
3 | 1, 2 | mpbir 145 | 1 ⊢ ∪ ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ⊆ wss 3000 ∅c0 3287 {csn 3450 ∪ cuni 3659 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-in 3006 df-ss 3013 df-nul 3288 df-sn 3456 df-uni 3660 |
This theorem is referenced by: iununir 3818 unixp0im 4980 iotanul 5008 1st0 5929 2nd0 5930 brtpos0 6031 tpostpos 6043 nnsucuniel 6270 sup00 6752 0opn 11759 nnpredcl 12156 |
Copyright terms: Public domain | W3C validator |