Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > bastop2 | Structured version Visualization version GIF version | ||
| Description: A version of bastop1 22847 that doesn't have π΅ β π½ in the antecedent. (Contributed by NM, 3-Feb-2008.) |
| Ref | Expression |
|---|---|
| bastop2 | β’ (π½ β Top β ((topGenβπ΅) = π½ β (π΅ β π½ β§ βπ₯ β π½ βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1 2815 | . . . . . . . 8 β’ ((topGenβπ΅) = π½ β ((topGenβπ΅) β Top β π½ β Top)) | |
| 2 | 1 | biimparc 479 | . . . . . . 7 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β (topGenβπ΅) β Top) |
| 3 | tgclb 22824 | . . . . . . 7 β’ (π΅ β TopBases β (topGenβπ΅) β Top) | |
| 4 | 2, 3 | sylibr 233 | . . . . . 6 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β π΅ β TopBases) |
| 5 | bastg 22820 | . . . . . 6 β’ (π΅ β TopBases β π΅ β (topGenβπ΅)) | |
| 6 | 4, 5 | syl 17 | . . . . 5 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β π΅ β (topGenβπ΅)) |
| 7 | simpr 484 | . . . . 5 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β (topGenβπ΅) = π½) | |
| 8 | 6, 7 | sseqtrd 4017 | . . . 4 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β π΅ β π½) |
| 9 | 8 | ex 412 | . . 3 β’ (π½ β Top β ((topGenβπ΅) = π½ β π΅ β π½)) |
| 10 | 9 | pm4.71rd 562 | . 2 β’ (π½ β Top β ((topGenβπ΅) = π½ β (π΅ β π½ β§ (topGenβπ΅) = π½))) |
| 11 | bastop1 22847 | . . 3 β’ ((π½ β Top β§ π΅ β π½) β ((topGenβπ΅) = π½ β βπ₯ β π½ βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦))) | |
| 12 | 11 | pm5.32da 578 | . 2 β’ (π½ β Top β ((π΅ β π½ β§ (topGenβπ΅) = π½) β (π΅ β π½ β§ βπ₯ β π½ βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦)))) |
| 13 | 10, 12 | bitrd 279 | 1 β’ (π½ β Top β ((topGenβπ΅) = π½ β (π΅ β π½ β§ βπ₯ β π½ βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 βwex 1773 β wcel 2098 βwral 3055 β wss 3943 βͺ cuni 4902 βcfv 6536 topGenctg 17390 Topctop 22746 TopBasesctb 22799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-topgen 17396 df-top 22747 df-bases 22800 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |