Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > uni0b | Unicode version |
Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.) |
Ref | Expression |
---|---|
uni0b |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3412 | . . . 4 | |
2 | 1 | ralbii 2463 | . . 3 |
3 | ralcom4 2734 | . . 3 | |
4 | 2, 3 | bitri 183 | . 2 |
5 | dfss3 3118 | . . 3 | |
6 | velsn 3577 | . . . 4 | |
7 | 6 | ralbii 2463 | . . 3 |
8 | 5, 7 | bitri 183 | . 2 |
9 | eluni2 3777 | . . . . 5 | |
10 | 9 | notbii 658 | . . . 4 |
11 | 10 | albii 1450 | . . 3 |
12 | eq0 3412 | . . 3 | |
13 | ralnex 2445 | . . . 4 | |
14 | 13 | albii 1450 | . . 3 |
15 | 11, 12, 14 | 3bitr4i 211 | . 2 |
16 | 4, 8, 15 | 3bitr4ri 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wal 1333 wceq 1335 wcel 2128 wral 2435 wrex 2436 wss 3102 c0 3394 csn 3560 cuni 3773 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 df-uni 3774 |
This theorem is referenced by: uni0c 3799 uni0 3800 |
Copyright terms: Public domain | W3C validator |