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

Theorem prmg 3704
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 3701 . 2 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 orc 707 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 3600 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 vex 2733 . . . . 5 𝑥 ∈ V
54elpr 3604 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 3, 53imtr4i 200 . . 3 (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
76eximi 1593 . 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 703   = wceq 1348  wex 1485  wcel 2141  {csn 3583  {cpr 3584
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590
This theorem is referenced by:  prm  3706  opm  4219  onintexmid  4557
  Copyright terms: Public domain W3C validator