Theorem indislem 20804

Description: A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
indislem	|- { (/) , ( _I ` A ) } = { (/) , A }

Proof of Theorem indislem

Step	Hyp	Ref	Expression
1		fvi 6255	. . 3 |- ( A e. _V -> ( _I ` A ) = A )
2	1	preq2d 4275	. 2 |- ( A e. _V -> { (/) , ( _I ` A ) } = { (/) , A } )
3		dfsn2 4190	. . . 4 |- { (/) } = { (/) , (/) }
4	3	eqcomi 2631	. . 3 |- { (/) , (/) } = { (/) }
5		fvprc 6185	. . . 4 |- ( -. A e. _V -> ( _I ` A ) = (/) )
6	5	preq2d 4275	. . 3 |- ( -. A e. _V -> { (/) , ( _I ` A ) } = { (/) , (/) } )
7		prprc2 4301	. . 3 |- ( -. A e. _V -> { (/) , A } = { (/) } )
8	4, 6, 7	3eqtr4a 2682	. 2 |- ( -. A e. _V -> { (/) , ( _I ` A ) } = { (/) , A } )
9	2, 8	pm2.61i 176	1 |- { (/) , ( _I ` A ) } = { (/) , A }

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   {cpr 4179   _I cid 5023   ` cfv 5888

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

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-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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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

This theorem is referenced by:  indistop  20806  indisuni  20807  indiscld  20895  indisconn  21221  txindis  21437  hmphindis  21600