Theorem hmphindis 21600

Description: Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
hmphdis.1	|- X = U. J

Assertion

Ref	Expression
hmphindis	|- ( J ~= { (/) , A } -> J = { (/) , X } )

Proof of Theorem hmphindis

Step	Hyp	Ref	Expression
1		dfsn2 4190	. . 3 |- { (/) } = { (/) , (/) }
2		indislem 20804	. . . . . . 7 |- { (/) , ( _I ` A ) } = { (/) , A }
3		preq2 4269	. . . . . . . 8 |- ( ( _I ` A ) = (/) -> { (/) , ( _I ` A ) } = { (/) , (/) } )
4	3, 1	syl6eqr 2674	. . . . . . 7 |- ( ( _I ` A ) = (/) -> { (/) , ( _I ` A ) } = { (/) } )
5	2, 4	syl5eqr 2670	. . . . . 6 |- ( ( _I ` A ) = (/) -> { (/) , A } = { (/) } )
6	5	breq2d 4665	. . . . 5 |- ( ( _I ` A ) = (/) -> ( J ~= { (/) , A } <-> J ~= { (/) } ) )
7	6	biimpac 503	. . . 4 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> J ~= { (/) } )
8		hmph0 21598	. . . 4 |- ( J ~= { (/) } <-> J = { (/) } )
9	7, 8	sylib 208	. . 3 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> J = { (/) } )
10	9	unieqd 4446	. . . . 5 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> U. J = U. { (/) } )
11		hmphdis.1	. . . . 5 |- X = U. J
12		0ex 4790	. . . . . . 7 |- (/) e. _V
13	12	unisn 4451	. . . . . 6 |- U. { (/) } = (/)
14	13	eqcomi 2631	. . . . 5 |- (/) = U. { (/) }
15	10, 11, 14	3eqtr4g 2681	. . . 4 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> X = (/) )
16	15	preq2d 4275	. . 3 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> { (/) , X } = { (/) , (/) } )
17	1, 9, 16	3eqtr4a 2682	. 2 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) = (/) ) -> J = { (/) , X } )
18		hmphen 21588	. . . . . 6 |- ( J ~= { (/) , A } -> J ~~ { (/) , A } )
19	18	adantr 481	. . . . 5 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J ~~ { (/) , A } )
20		necom 2847	. . . . . . . 8 |- ( ( _I ` A ) =/= (/) <-> (/) =/= ( _I ` A ) )
21		fvex 6201	. . . . . . . . 9 |- ( _I ` A ) e. _V
22		pr2nelem 8827	. . . . . . . . 9 |- ( ( (/) e. _V /\ ( _I ` A ) e. _V /\ (/) =/= ( _I ` A ) ) -> { (/) , ( _I ` A ) } ~~ 2o )
23	12, 21, 22	mp3an12 1414	. . . . . . . 8 |- ( (/) =/= ( _I ` A ) -> { (/) , ( _I ` A ) } ~~ 2o )
24	20, 23	sylbi 207	. . . . . . 7 |- ( ( _I ` A ) =/= (/) -> { (/) , ( _I ` A ) } ~~ 2o )
25	24	adantl 482	. . . . . 6 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> { (/) , ( _I ` A ) } ~~ 2o )
26	2, 25	syl5eqbrr 4689	. . . . 5 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> { (/) , A } ~~ 2o )
27		entr 8008	. . . . 5 |- ( ( J ~~ { (/) , A } /\ { (/) , A } ~~ 2o ) -> J ~~ 2o )
28	19, 26, 27	syl2anc 693	. . . 4 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J ~~ 2o )
29		hmphtop1 21582	. . . . . . 7 |- ( J ~= { (/) , A } -> J e. Top )
30	29	adantr 481	. . . . . 6 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J e. Top )
31	11	toptopon 20722	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` X ) )
32	30, 31	sylib 208	. . . . 5 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J e. (TopOn ` X ) )
33		en2top 20789	. . . . 5 |- ( J e. (TopOn ` X ) -> ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) )
34	32, 33	syl 17	. . . 4 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> ( J ~~ 2o <-> ( J = { (/) , X } /\ X =/= (/) ) ) )
35	28, 34	mpbid 222	. . 3 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> ( J = { (/) , X } /\ X =/= (/) ) )
36	35	simpld 475	. 2 |- ( ( J ~= { (/) , A } /\ ( _I ` A ) =/= (/) ) -> J = { (/) , X } )
37	17, 36	pm2.61dane 2881	1 |- ( J ~= { (/) , A } -> J = { (/) , X } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   {cpr 4179   U.cuni 4436   class class class wbr 4653   _I cid 5023   ` cfv 5888   2oc2o 7554   ~~ cen 7952   Topctop 20698  TopOnctopon 20715   ~= chmph 21557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558  df-hmph 21559

This theorem is referenced by: (None)