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Mirrors > Home > ILE Home > Th. List > unexb | Unicode version |
Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
Ref | Expression |
---|---|
unexb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3148 |
. . . 4
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2 | 1 | eleq1d 2157 |
. . 3
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3 | uneq2 3149 |
. . . 4
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4 | 3 | eleq1d 2157 |
. . 3
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5 | vex 2623 |
. . . 4
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6 | vex 2623 |
. . . 4
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7 | 5, 6 | unex 4276 |
. . 3
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8 | 2, 4, 7 | vtocl2g 2684 |
. 2
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9 | ssun1 3164 |
. . . 4
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10 | ssexg 3984 |
. . . 4
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11 | 9, 10 | mpan 416 |
. . 3
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12 | ssun2 3165 |
. . . 4
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13 | ssexg 3984 |
. . . 4
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14 | 12, 13 | mpan 416 |
. . 3
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15 | 11, 14 | jca 301 |
. 2
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16 | 8, 15 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 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-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-uni 3660 |
This theorem is referenced by: unexg 4278 sucexb 4327 frecabex 6177 |
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