Theorem prnz 3796

Description: A pair containing a set is not empty. It is also inhabited (see prm 3797). (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 3778	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		ne0i 3500	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
4	2, 3	ax-mp 5	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2201   ≠ wne 2401  Vcvv 2801  ∅c0 3493  {cpr 3671

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-v 2803  df-dif 3201  df-un 3203  df-nul 3494  df-sn 3676  df-pr 3677

This theorem is referenced by:  prnzg  3798