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Theorem prnzg 3730
Description: A pair containing a set is not empty. It is also inhabited (see prmg 3727). (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
prnzg (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Proof of Theorem prnzg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 preq1 3683 . . 3 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
21neeq1d 2377 . 2 (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3 vex 2754 . . 3 𝑥 ∈ V
43prnz 3728 . 2 {𝑥, 𝐵} ≠ ∅
52, 4vtoclg 2811 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2159  wne 2359  c0 3436  {cpr 3607
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-in1 615  ax-in2 616  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 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-v 2753  df-dif 3145  df-un 3147  df-nul 3437  df-sn 3612  df-pr 3613
This theorem is referenced by:  0nelop  4262
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