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

Theorem prmg 3639
 Description: A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
prmg (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem prmg
StepHypRef Expression
1 snmg 3636 . 2 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 orc 701 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 3539 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 vex 2684 . . . . 5 𝑥 ∈ V
54elpr 3543 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 3, 53imtr4i 200 . . 3 (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
76eximi 1579 . 2 (∃𝑥 𝑥 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
81, 7syl 14 1 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {csn 3522  {cpr 3523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529 This theorem is referenced by:  prm  3641  opm  4151  onintexmid  4482
 Copyright terms: Public domain W3C validator