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Mirrors > Home > MPE Home > Th. List > bastop2 | Structured version Visualization version GIF version |
Description: A version of bastop1 22359 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 22336 | . . . . . . 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 22332 | . . . . . 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 3985 | . . . 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 22359 | . . 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 3061 β wss 3911 βͺ cuni 4866 βcfv 6497 topGenctg 17324 Topctop 22258 TopBasesctb 22311 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-topgen 17330 df-top 22259 df-bases 22312 |
This theorem is referenced by: (None) |
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