ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tpid2g GIF version

Theorem tpid2g 3728
Description: Closed theorem form of tpid2 3727. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid2g (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})

Proof of Theorem tpid2g
StepHypRef Expression
1 eqid 2189 . . 3 𝐴 = 𝐴
213mix2i 1172 . 2 (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)
3 eltpg 3659 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)))
42, 3mpbiri 168 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 979   = wceq 1364  wcel 2160  {ctp 3616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2758  df-un 3153  df-sn 3620  df-pr 3621  df-tp 3622
This theorem is referenced by:  rngplusgg  12728  srngplusgd  12739  lmodplusgd  12757  ipsaddgd  12769  ipsvscad  12772  topgrpplusgd  12789
  Copyright terms: Public domain W3C validator