Theorem prnz 4310

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 4297	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 3923	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200  ∅c0 3915  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  prnzgOLD  4312  opnz  4942  propssopi  4971  fiint  8237  wilthlem2  24795  upgrbi  25988  wlkvtxiedg  26520  shincli  28221  chincli  28319  spr0nelg  41726  sprvalpwn0  41733