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Theorem prnz 4453
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 𝐴 ∈ V
Assertion
Ref Expression
prnz {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 𝐴 ∈ V
21prid1 4441 . 2 𝐴 ∈ {𝐴, 𝐵}
32ne0ii 4066 1 {𝐴, 𝐵} ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  wne 2932  Vcvv 3340  c0 4058  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324
This theorem is referenced by:  prnzgOLD  4455  opnz  5090  propssopi  5119  fiint  8402  wilthlem2  24994  upgrbi  26187  wlkvtxiedg  26730  shincli  28530  chincli  28628  spr0nelg  42236  sprvalpwn0  42243
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