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Mirrors > Home > ILE Home > Th. List > basis1 | Unicode version |
Description: Property of a basis. (Contributed by NM, 16-Jul-2006.) |
Ref | Expression |
---|---|
basis1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isbasisg 12757 | . . . 4 | |
2 | 1 | ibi 175 | . . 3 |
3 | ineq1 3321 | . . . . 5 | |
4 | 3 | pweqd 3569 | . . . . . . 7 |
5 | 4 | ineq2d 3328 | . . . . . 6 |
6 | 5 | unieqd 3805 | . . . . 5 |
7 | 3, 6 | sseq12d 3178 | . . . 4 |
8 | ineq2 3322 | . . . . 5 | |
9 | 8 | pweqd 3569 | . . . . . . 7 |
10 | 9 | ineq2d 3328 | . . . . . 6 |
11 | 10 | unieqd 3805 | . . . . 5 |
12 | 8, 11 | sseq12d 3178 | . . . 4 |
13 | 7, 12 | rspc2v 2847 | . . 3 |
14 | 2, 13 | syl5com 29 | . 2 |
15 | 14 | 3impib 1196 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 wral 2448 cin 3120 wss 3121 cpw 3564 cuni 3794 ctb 12755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 df-uni 3795 df-bases 12756 |
This theorem is referenced by: (None) |
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