Theorem topgele 14703

Description: The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
topgele	|- ( J e. (TopOn ` X ) -> ( { (/) , X } C_ J /\ J C_ ~P X ) )

Proof of Theorem topgele

Step	Hyp	Ref	Expression
1		topontop 14688	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2		0opn 14680	. . . 4 |- ( J e. Top -> (/) e. J )
3	1, 2	syl 14	. . 3 |- ( J e. (TopOn ` X ) -> (/) e. J )
4		toponmax 14699	. . 3 |- ( J e. (TopOn ` X ) -> X e. J )
5		0ex 4211	. . . 4 |- (/) e. _V
6		prssg 3825	. . . 4 |- ( ( (/) e. _V /\ X e. J ) -> ( ( (/) e. J /\ X e. J ) <-> { (/) , X } C_ J ) )
7	5, 4, 6	sylancr 414	. . 3 |- ( J e. (TopOn ` X ) -> ( ( (/) e. J /\ X e. J ) <-> { (/) , X } C_ J ) )
8	3, 4, 7	mpbi2and 949	. 2 |- ( J e. (TopOn ` X ) -> { (/) , X } C_ J )
9		toponuni 14689	. . . 4 |- ( J e. (TopOn ` X ) -> X = U. J )
10		eqimss2 3279	. . . 4 |- ( X = U. J -> U. J C_ X )
11	9, 10	syl 14	. . 3 |- ( J e. (TopOn ` X ) -> U. J C_ X )
12		sspwuni 4050	. . 3 |- ( J C_ ~P X <-> U. J C_ X )
13	11, 12	sylibr 134	. 2 |- ( J e. (TopOn ` X ) -> J C_ ~P X )
14	8, 13	jca 306	1 |- ( J e. (TopOn ` X ) -> ( { (/) , X } C_ J /\ J C_ ~P X ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   (/)c0 3491   ~Pcpw 3649   {cpr 3667   U.cuni 3888   ` cfv 5318   Topctop 14671  TopOnctopon 14684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-top 14672  df-topon 14685

This theorem is referenced by: (None)