Theorem dissneq 33188

Description: Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)

Hypothesis

Ref	Expression
dissneq.c	|- C = { u | E. x e. A u = { x } }

Assertion

Ref	Expression
dissneq	|- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> B = ~P A )

Distinct variable group:   u, A, x

Allowed substitution hints:   B( x, u)   C( x, u)

Proof of Theorem dissneq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dissneq.c	. . 3 |- C = { u | E. x e. A u = { x } }
2		sneq 4187	. . . . . 6 |- ( z = x -> { z } = { x } )
3	2	eqeq2d 2632	. . . . 5 |- ( z = x -> ( u = { z } <-> u = { x } ) )
4	3	cbvrexv 3172	. . . 4 |- ( E. z e. A u = { z } <-> E. x e. A u = { x } )
5	4	abbii 2739	. . 3 |- { u | E. z e. A u = { z } } = { u | E. x e. A u = { x } }
6	1, 5	eqtr4i 2647	. 2 |- C = { u | E. z e. A u = { z } }
7	6	dissneqlem 33187	1 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> B = ~P A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   {csn 4177   ` cfv 5888  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-top 20699  df-topon 20716

This theorem is referenced by:  topdifinffinlem  33195