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Mathbox for Jeff Hankins |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isfne4b | Structured version Visualization version GIF version |
Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
isfne.1 | β’ π = βͺ π΄ |
isfne.2 | β’ π = βͺ π΅ |
Ref | Expression |
---|---|
isfne4b | β’ (π΅ β π β (π΄Fneπ΅ β (π = π β§ (topGenβπ΄) β (topGenβπ΅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfne.1 | . . 3 β’ π = βͺ π΄ | |
2 | isfne.2 | . . 3 β’ π = βͺ π΅ | |
3 | 1, 2 | isfne4 35732 | . 2 β’ (π΄Fneπ΅ β (π = π β§ π΄ β (topGenβπ΅))) |
4 | simpr 484 | . . . . . . 7 β’ ((π΅ β π β§ π = π) β π = π) | |
5 | 4, 1, 2 | 3eqtr3g 2789 | . . . . . 6 β’ ((π΅ β π β§ π = π) β βͺ π΄ = βͺ π΅) |
6 | uniexg 7726 | . . . . . . 7 β’ (π΅ β π β βͺ π΅ β V) | |
7 | 6 | adantr 480 | . . . . . 6 β’ ((π΅ β π β§ π = π) β βͺ π΅ β V) |
8 | 5, 7 | eqeltrd 2827 | . . . . 5 β’ ((π΅ β π β§ π = π) β βͺ π΄ β V) |
9 | uniexb 7747 | . . . . 5 β’ (π΄ β V β βͺ π΄ β V) | |
10 | 8, 9 | sylibr 233 | . . . 4 β’ ((π΅ β π β§ π = π) β π΄ β V) |
11 | simpl 482 | . . . 4 β’ ((π΅ β π β§ π = π) β π΅ β π) | |
12 | tgss3 22839 | . . . 4 β’ ((π΄ β V β§ π΅ β π) β ((topGenβπ΄) β (topGenβπ΅) β π΄ β (topGenβπ΅))) | |
13 | 10, 11, 12 | syl2anc 583 | . . 3 β’ ((π΅ β π β§ π = π) β ((topGenβπ΄) β (topGenβπ΅) β π΄ β (topGenβπ΅))) |
14 | 13 | pm5.32da 578 | . 2 β’ (π΅ β π β ((π = π β§ (topGenβπ΄) β (topGenβπ΅)) β (π = π β§ π΄ β (topGenβπ΅)))) |
15 | 3, 14 | bitr4id 290 | 1 β’ (π΅ β π β (π΄Fneπ΅ β (π = π β§ (topGenβπ΄) β (topGenβπ΅)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 βͺ cuni 4902 class class class wbr 5141 βcfv 6536 topGenctg 17389 Fnecfne 35728 |
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-iun 4992 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 17395 df-fne 35729 |
This theorem is referenced by: fnetr 35743 fneval 35744 |
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