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Theorem prmg 3561
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 3558 . 2 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 orc 668 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 3463 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 vex 2622 . . . . 5 𝑥 ∈ V
54elpr 3467 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 3, 53imtr4i 199 . . 3 (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
76eximi 1536 . 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 664   = wceq 1289  wex 1426  wcel 1438  {csn 3446  {cpr 3447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453
This theorem is referenced by:  prm  3563  opm  4061  onintexmid  4388
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