Theorem indislem 22944

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 6910	. . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
2	1	preq2d 4697	. 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3		dfsn2 4593	. . . 4 ⊢ {∅} = {∅, ∅}
4	3	eqcomi 2745	. . 3 ⊢ {∅, ∅} = {∅}
5		fvprc 6826	. . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
6	5	preq2d 4697	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7		prprc2 4723	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
8	4, 6, 7	3eqtr4a 2797	. 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
9	2, 8	pm2.61i 182	1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  {csn 4580  {cpr 4582   I cid 5518  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  indistop  22946  indisuni  22947  indiscld  23035  indisconn  23362  txindis  23578  hmphindis  23741