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Mirrors > Home > ILE Home > Th. List > un00 | Unicode version |
Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.) |
Ref | Expression |
---|---|
un00 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq12 3195 | . . 3 | |
2 | un0 3366 | . . 3 | |
3 | 1, 2 | syl6eq 2166 | . 2 |
4 | ssun1 3209 | . . . . 5 | |
5 | sseq2 3091 | . . . . 5 | |
6 | 4, 5 | mpbii 147 | . . . 4 |
7 | ss0b 3372 | . . . 4 | |
8 | 6, 7 | sylib 121 | . . 3 |
9 | ssun2 3210 | . . . . 5 | |
10 | sseq2 3091 | . . . . 5 | |
11 | 9, 10 | mpbii 147 | . . . 4 |
12 | ss0b 3372 | . . . 4 | |
13 | 11, 12 | sylib 121 | . . 3 |
14 | 8, 13 | jca 304 | . 2 |
15 | 3, 14 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 cun 3039 wss 3041 c0 3333 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 |
This theorem is referenced by: undisj1 3390 undisj2 3391 disjpr2 3557 |
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