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Theorem prnzgOLD 4255
Description: Obsolete proof of prnzg 4254 as of 23-Jul-2021. (Contributed by FL, 19-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
prnzgOLD (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Proof of Theorem prnzgOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 preq1 4212 . . 3 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
21neeq1d 2841 . 2 (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3 vex 3176 . . 3 𝑥 ∈ V
43prnz 4253 . 2 {𝑥, 𝐵} ≠ ∅
52, 4vtoclg 3239 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wne 2780  c0 3874  {cpr 4127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128
This theorem is referenced by: (None)
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