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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 3319 |
. . . 4
| |
| 2 | 1 | eleq1d 2273 |
. . 3
|
| 3 | uneq2 3320 |
. . . 4
| |
| 4 | 3 | eleq1d 2273 |
. . 3
|
| 5 | vex 2774 |
. . . 4
| |
| 6 | vex 2774 |
. . . 4
| |
| 7 | 5, 6 | unex 4487 |
. . 3
|
| 8 | 2, 4, 7 | vtocl2g 2836 |
. 2
|
| 9 | ssun1 3335 |
. . . 4
| |
| 10 | ssexg 4182 |
. . . 4
| |
| 11 | 9, 10 | mpan 424 |
. . 3
|
| 12 | ssun2 3336 |
. . . 4
| |
| 13 | ssexg 4182 |
. . . 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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pr 4252 ax-un 4479 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-sn 3638 df-pr 3639 df-uni 3850 |
| This theorem is referenced by: unexg 4489 sucexb 4544 frecabex 6483 djuexb 7145 |
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