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Theorem prnzgOLD 4287
 Description: Obsolete proof of prnzg 4286 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 4243 . . 3 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
21neeq1d 2855 . 2 (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3 vex 3194 . . 3 𝑥 ∈ V
43prnz 4285 . 2 {𝑥, 𝐵} ≠ ∅
52, 4vtoclg 3257 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1992   ≠ wne 2796  ∅c0 3896  {cpr 4155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-v 3193  df-dif 3563  df-un 3565  df-nul 3897  df-sn 4154  df-pr 4156 This theorem is referenced by: (None)
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