Theorem topgele 14894

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 14879	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2		0opn 14871	. . . 4 |- ( J e. Top -> (/) e. J )
3	1, 2	syl 14	. . 3 |- ( J e. (TopOn ` X ) -> (/) e. J )
4		toponmax 14890	. . 3 |- ( J e. (TopOn ` X ) -> X e. J )
5		0ex 4237	. . . 4 |- (/) e. _V
6		prssg 3851	. . . 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 952	. 2 |- ( J e. (TopOn ` X ) -> { (/) , X } C_ J )
9		toponuni 14880	. . . 4 |- ( J e. (TopOn ` X ) -> X = U. J )
10		eqimss2 3293	. . . 4 |- ( X = U. J -> U. J C_ X )
11	9, 10	syl 14	. . 3 |- ( J e. (TopOn ` X ) -> U. J C_ X )
12		sspwuni 4076	. . 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 1398   e. wcel 2203   _Vcvv 2813   C_ wss 3211   (/)c0 3508   ~Pcpw 3669   {cpr 3690   U.cuni 3914   ` cfv 5352   Topctop 14862  TopOnctopon 14875

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-top 14863  df-topon 14876

This theorem is referenced by: (None)