MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indislem Structured version   Visualization version   GIF version

Theorem indislem 20744
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
StepHypRef Expression
1 fvi 6222 . . 3 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
21preq2d 4252 . 2 (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3 dfsn2 4168 . . . 4 {∅} = {∅, ∅}
43eqcomi 2630 . . 3 {∅, ∅} = {∅}
5 fvprc 6152 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
65preq2d 4252 . . 3 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7 prprc2 4278 . . 3 𝐴 ∈ V → {∅, 𝐴} = {∅})
84, 6, 73eqtr4a 2681 . 2 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
92, 8pm2.61i 176 1 {∅, ( I ‘𝐴)} = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3190  c0 3897  {csn 4155  {cpr 4157   I cid 4994  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  indistop  20746  indisuni  20747  indiscld  20835  indisconn  21161  txindis  21377  hmphindis  21540
  Copyright terms: Public domain W3C validator