Theorem indislem 23225

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 6954	. . 3 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
2	1	preq2d 4701	. 2 ⊢ (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3		dfsn2 4597	. . . 4 ⊢ {∅} = {∅, ∅}
4	3	eqcomi 2769	. . 3 ⊢ {∅, ∅} = {∅}
5		fvprc 6870	. . . 4 ⊢ (¬ 𝐴 ∈ V → ( I ‘𝐴) = ∅)
6	5	preq2d 4701	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7		prprc2 4727	. . 3 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
8	4, 6, 7	3eqtr4a 2821	. 2 ⊢ (¬ 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
9	2, 8	pm2.61i 184	1 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  {csn 4584  {cpr 4586   I cid 5549  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541

This theorem is used by:  indistop  23227  indisuni  23228  indiscld  23316  indisconn  23643  txindis  23860  hmphindis  24023