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Theorem prnzg 3617
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 3570 . . 3 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
21neeq1d 2303 . 2 (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3 vex 2663 . . 3 𝑥 ∈ V
43prnz 3615 . 2 {𝑥, 𝐵} ≠ ∅
52, 4vtoclg 2720 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  wne 2285  c0 3333  {cpr 3498
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-dif 3043  df-un 3045  df-nul 3334  df-sn 3503  df-pr 3504
This theorem is referenced by:  0nelop  4140
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