Theorem prnz 4416

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 4404	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4031	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2103   ≠ wne 2896  Vcvv 3304  ∅c0 4023  {cpr 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-v 3306  df-dif 3683  df-un 3685  df-nul 4024  df-sn 4286  df-pr 4288

This theorem is referenced by:  prnzgOLD  4418  opnz  5046  propssopi  5075  fiint  8353  wilthlem2  24915  upgrbi  26108  wlkvtxiedg  26651  shincli  28451  chincli  28549  spr0nelg  42153  sprvalpwn0  42160