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Theorem prnzg 3651
Description: A pair containing a set is not empty. (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 3604 . . 3 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
21neeq1d 2327 . 2 (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3 vex 2690 . . 3 𝑥 ∈ V
43prnz 3649 . 2 {𝑥, 𝐵} ≠ ∅
52, 4vtoclg 2747 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481  wne 2309  c0 3364  {cpr 3529
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2689  df-dif 3074  df-un 3076  df-nul 3365  df-sn 3534  df-pr 3535
This theorem is referenced by:  0nelop  4174
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