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Theorem prmg 3725
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 3722 . 2 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 orc 713 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 3621 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 vex 2752 . . . . 5 𝑥 ∈ V
54elpr 3625 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 3, 53imtr4i 201 . . 3 (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
76eximi 1610 . 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 709   = wceq 1363  wex 1502  wcel 2158  {csn 3604  {cpr 3605
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611
This theorem is referenced by:  prm  3727  opm  4246  onintexmid  4584  subrngin  13490  subrgin  13521  lssincl  13631
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