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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 3297 |
. . . 4
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2 | 1 | eleq1d 2258 |
. . 3
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3 | uneq2 3298 |
. . . 4
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4 | 3 | eleq1d 2258 |
. . 3
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5 | vex 2755 |
. . . 4
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6 | vex 2755 |
. . . 4
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7 | 5, 6 | unex 4459 |
. . 3
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8 | 2, 4, 7 | vtocl2g 2816 |
. 2
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9 | ssun1 3313 |
. . . 4
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10 | ssexg 4157 |
. . . 4
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11 | 9, 10 | mpan 424 |
. . 3
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12 | ssun2 3314 |
. . . 4
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13 | ssexg 4157 |
. . . 4
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14 | 12, 13 | mpan 424 |
. . 3
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15 | 11, 14 | jca 306 |
. 2
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16 | 8, 15 | impbii 126 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-uni 3825 |
This theorem is referenced by: unexg 4461 sucexb 4514 frecabex 6423 djuexb 7073 |
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