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Theorem prmg 3517
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 3514 . 2 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 orc 643 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝐵))
3 velsn 3420 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 vex 2577 . . . . 5 𝑥 ∈ V
54elpr 3424 . . . 4 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
62, 3, 53imtr4i 194 . . 3 (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
76eximi 1507 . 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 639   = wceq 1259  wex 1397  wcel 1409  {csn 3403  {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410
This theorem is referenced by:  prm  3519  opm  3999  onintexmid  4325
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