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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 3376 | . . . 4 | |
2 | 1 | ralbii 2439 | . . 3 |
3 | ralcom4 2703 | . . 3 | |
4 | 2, 3 | bitri 183 | . 2 |
5 | dfss3 3082 | . . 3 | |
6 | velsn 3539 | . . . 4 | |
7 | 6 | ralbii 2439 | . . 3 |
8 | 5, 7 | bitri 183 | . 2 |
9 | eluni2 3735 | . . . . 5 | |
10 | 9 | notbii 657 | . . . 4 |
11 | 10 | albii 1446 | . . 3 |
12 | eq0 3376 | . . 3 | |
13 | ralnex 2424 | . . . 4 | |
14 | 13 | albii 1446 | . . 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 1329 wceq 1331 wcel 1480 wral 2414 wrex 2415 wss 3066 c0 3358 csn 3522 cuni 3731 |
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-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-uni 3732 |
This theorem is referenced by: uni0c 3757 uni0 3758 |
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