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| Mirrors > Home > MPE Home > Th. List > bastop2 | Structured version Visualization version GIF version | ||
| Description: A version of bastop1 22496 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 2822 | . . . . . . . 8 β’ ((topGenβπ΅) = π½ β ((topGenβπ΅) β Top β π½ β Top)) | |
| 2 | 1 | biimparc 481 | . . . . . . 7 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β (topGenβπ΅) β Top) |
| 3 | tgclb 22473 | . . . . . . 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 22469 | . . . . . 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 486 | . . . . 5 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β (topGenβπ΅) = π½) | |
| 8 | 6, 7 | sseqtrd 4023 | . . . 4 β’ ((π½ β Top β§ (topGenβπ΅) = π½) β π΅ β π½) |
| 9 | 8 | ex 414 | . . 3 β’ (π½ β Top β ((topGenβπ΅) = π½ β π΅ β π½)) |
| 10 | 9 | pm4.71rd 564 | . 2 β’ (π½ β Top β ((topGenβπ΅) = π½ β (π΅ β π½ β§ (topGenβπ΅) = π½))) |
| 11 | bastop1 22496 | . . 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 580 | . 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 397 = wceq 1542 βwex 1782 β wcel 2107 βwral 3062 β wss 3949 βͺ cuni 4909 βcfv 6544 topGenctg 17383 Topctop 22395 TopBasesctb 22448 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-topgen 17389 df-top 22396 df-bases 22449 |
| This theorem is referenced by: (None) |
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