Theorem prnz 3769

Description: A pair containing a set is not empty. It is also inhabited (see prm 3770). (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 3752	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		ne0i 3478	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
4	2, 3	ax-mp 5	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2180   ≠ wne 2380  Vcvv 2779  ∅c0 3471  {cpr 3647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-v 2781  df-dif 3179  df-un 3181  df-nul 3472  df-sn 3652  df-pr 3653

This theorem is referenced by:  prnzg  3771