Theorem indislem 22956

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 6918	. . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
2	1	preq2d 4699	. 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3		dfsn2 4595	. . . 4 ⊢ {∅} = {∅, ∅}
4	3	eqcomi 2746	. . 3 ⊢ {∅, ∅} = {∅}
5		fvprc 6834	. . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
6	5	preq2d 4699	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7		prprc2 4725	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
8	4, 6, 7	3eqtr4a 2798	. 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
9	2, 8	pm2.61i 182	1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  {csn 4582  {cpr 4584   I cid 5526  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508

This theorem is referenced by:  indistop  22958  indisuni  22959  indiscld  23047  indisconn  23374  txindis  23590  hmphindis  23753