Theorem indislem 23008

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 6984	. . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
2	1	preq2d 4739	. 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3		dfsn2 4638	. . . 4 ⊢ {∅} = {∅, ∅}
4	3	eqcomi 2745	. . 3 ⊢ {∅, ∅} = {∅}
5		fvprc 6897	. . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
6	5	preq2d 4739	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7		prprc2 4765	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
8	4, 6, 7	3eqtr4a 2802	. 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
9	2, 8	pm2.61i 182	1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2107  Vcvv 3479  ∅c0 4332  {csn 4625  {cpr 4627   I cid 5576  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568

This theorem is referenced by:  indistop  23010  indisuni  23011  indiscld  23100  indisconn  23427  txindis  23643  hmphindis  23806