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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq12 3149 |
. . 3
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2 | un0 3316 |
. . 3
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3 | 1, 2 | syl6eq 2136 |
. 2
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4 | ssun1 3163 |
. . . . 5
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5 | sseq2 3048 |
. . . . 5
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6 | 4, 5 | mpbii 146 |
. . . 4
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7 | ss0b 3322 |
. . . 4
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8 | 6, 7 | sylib 120 |
. . 3
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9 | ssun2 3164 |
. . . . 5
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10 | sseq2 3048 |
. . . . 5
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11 | 9, 10 | mpbii 146 |
. . . 4
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12 | ss0b 3322 |
. . . 4
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13 | 11, 12 | sylib 120 |
. . 3
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14 | 8, 13 | jca 300 |
. 2
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15 | 3, 14 | impbii 124 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 |
This theorem is referenced by: undisj1 3340 undisj2 3341 disjpr2 3506 |
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