Theorem indislem 22058

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 6826	. . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
2	1	preq2d 4673	. 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3		dfsn2 4571	. . . 4 ⊢ {∅} = {∅, ∅}
4	3	eqcomi 2747	. . 3 ⊢ {∅, ∅} = {∅}
5		fvprc 6748	. . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
6	5	preq2d 4673	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7		prprc2 4699	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
8	4, 6, 7	3eqtr4a 2805	. 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
9	2, 8	pm2.61i 182	1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  {csn 4558  {cpr 4560   I cid 5479  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  indistop  22060  indisuni  22061  indiscld  22150  indisconn  22477  txindis  22693  hmphindis  22856