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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 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uneq1 3320 |
. . . 4
| |
| 2 | 1 | eleq1d 2274 |
. . 3
|
| 3 | uneq2 3321 |
. . . 4
| |
| 4 | 3 | eleq1d 2274 |
. . 3
|
| 5 | vex 2775 |
. . . 4
| |
| 6 | vex 2775 |
. . . 4
| |
| 7 | 5, 6 | unex 4488 |
. . 3
|
| 8 | 2, 4, 7 | vtocl2g 2837 |
. 2
|
| 9 | ssun1 3336 |
. . . 4
| |
| 10 | ssexg 4183 |
. . . 4
| |
| 11 | 9, 10 | mpan 424 |
. . 3
|
| 12 | ssun2 3337 |
. . . 4
| |
| 13 | ssexg 4183 |
. . . 4
| |
| 14 | 12, 13 | mpan 424 |
. . 3
|
| 15 | 11, 14 | jca 306 |
. 2
|
| 16 | 8, 15 | impbii 126 |
1
|
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pr 4253 ax-un 4480 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-sn 3639 df-pr 3640 df-uni 3851 |
| This theorem is referenced by: unexg 4490 sucexb 4545 frecabex 6484 djuexb 7146 |
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