Theorem prnz 4253

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

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4241	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 3882	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅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:  prnzgOLD  4255  opnz  4868  propssopi  4896  fiint  8122  wilthlem2  24595  upgrbi  25760  edgwlk  26059  umgrabi  26510  shincli  27605  chincli  27703  1wlkvtxiedg  40829