Theorem topgele 20734

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 20718	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2		0opn 20709	. . . 4 |- ( J e. Top -> (/) e. J )
3	1, 2	syl 17	. . 3 |- ( J e. (TopOn ` X ) -> (/) e. J )
4		toponmax 20730	. . 3 |- ( J e. (TopOn ` X ) -> X e. J )
5		0ex 4790	. . . 4 |- (/) e. _V
6		prssg 4350	. . . 4 |- ( ( (/) e. _V /\ X e. J ) -> ( ( (/) e. J /\ X e. J ) <-> { (/) , X } C_ J ) )
7	5, 4, 6	sylancr 695	. . 3 |- ( J e. (TopOn ` X ) -> ( ( (/) e. J /\ X e. J ) <-> { (/) , X } C_ J ) )
8	3, 4, 7	mpbi2and 956	. 2 |- ( J e. (TopOn ` X ) -> { (/) , X } C_ J )
9		toponuni 20719	. . . 4 |- ( J e. (TopOn ` X ) -> X = U. J )
10		eqimss2 3658	. . . 4 |- ( X = U. J -> U. J C_ X )
11	9, 10	syl 17	. . 3 |- ( J e. (TopOn ` X ) -> U. J C_ X )
12		sspwuni 4611	. . 3 |- ( J C_ ~P X <-> U. J C_ X )
13	11, 12	sylibr 224	. 2 |- ( J e. (TopOn ` X ) -> J C_ ~P X )
14	8, 13	jca 554	1 |- ( J e. (TopOn ` X ) -> ( { (/) , X } C_ J /\ J C_ ~P X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {cpr 4179   U.cuni 4436   ` cfv 5888   Topctop 20698  TopOnctopon 20715

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-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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-topon 20716

This theorem is referenced by:  topsn  20735  txindis  21437  dissneqlem  33187  ntrf2  38422