Theorem sbtpsines 14905

Description: The only subspace topology induced by the topology {(/)}.

Assertion

Ref	Expression
sbtpsines	|- (A e. B -> (subSp` <.A, {(/)}>.) = {(/)})

Proof of Theorem sbtpsines

Step	Hyp	Ref	Expression
1		elisset 2299	. . . 4 |- (A e. B -> A e. _V)
2		sn0top 8917	. . . . 5 |- {(/)} e. Top
3		visset 2295	. . . . 5 |- x e. _V
4		issubspt 10247	. . . . 5 |- (({(/)} e. Top /\ x e. _V /\ A e. _V) -> (x e. (subSp` <.A, {(/)}>.) <-> E.y e. {(/)}x = (y i^i A)))
5	2, 3, 4	mp3an12 1181	. . . 4 |- (A e. _V -> (x e. (subSp` <.A, {(/)}>.) <-> E.y e. {(/)}x = (y i^i A)))
6	1, 5	syl 12	. . 3 |- (A e. B -> (x e. (subSp` <.A, {(/)}>.) <-> E.y e. {(/)}x = (y i^i A)))
7		df-rex 2110	. . . . 5 |- (E.y e. {(/)}x = (y i^i A) <-> E.y(y e. {(/)} /\ x = (y i^i A)))
8		elsn 3058	. . . . . . 7 |- (y e. {(/)} <-> y = (/))
9	8	anbi1i 539	. . . . . 6 |- ((y e. {(/)} /\ x = (y i^i A)) <-> (y = (/) /\ x = (y i^i A)))
10	9	exbii 1398	. . . . 5 |- (E.y(y e. {(/)} /\ x = (y i^i A)) <-> E.y(y = (/) /\ x = (y i^i A)))
11		0ex 3446	. . . . . 6 |- (/) e. _V
12		ineq1 2789	. . . . . . . 8 |- (y = (/) -> (y i^i A) = ((/) i^i A))
13	12	eqeq2d 1895	. . . . . . 7 |- (y = (/) -> (x = (y i^i A) <-> x = ((/) i^i A)))
14		incom 2787	. . . . . . . . 9 |- ((/) i^i A) = (A i^i (/))
15		in0 2897	. . . . . . . . 9 |- (A i^i (/)) = (/)
16	14, 15	eqtri 1908	. . . . . . . 8 |- ((/) i^i A) = (/)
17	16	eqeq2i 1894	. . . . . . 7 |- (x = ((/) i^i A) <-> x = (/))
18	13, 17	syl6bb 595	. . . . . 6 |- (y = (/) -> (x = (y i^i A) <-> x = (/)))
19	11, 18	ceqsexv 2325	. . . . 5 |- (E.y(y = (/) /\ x = (y i^i A)) <-> x = (/))
20	7, 10, 19	3bitri 194	. . . 4 |- (E.y e. {(/)}x = (y i^i A) <-> x = (/))
21		elsn 3058	. . . 4 |- (x e. {(/)} <-> x = (/))
22	20, 21	bitr4i 193	. . 3 |- (E.y e. {(/)}x = (y i^i A) <-> x e. {(/)})
23	6, 22	syl6bb 595	. 2 |- (A e. B -> (x e. (subSp` <.A, {(/)}>.) <-> x e. {(/)}))
24	23	eqrdv 1882	1 |- (A e. B -> (subSp` <.A, {(/)}>.) = {(/)})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292   i^i cin 2592  (/)c0 2875  {csn 3044  <.cop 3046  ` cfv 3998  Topctop 8857  subSpcsubsp 10242

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-top 8861  df-subsp 10243

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