Theorem prm 3699

Description: A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prm	⊢ ∃𝑥 𝑥 ∈ {𝐴, 𝐵}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem prm

Step	Hyp	Ref	Expression
1		prnz.1	. 2 ⊢ 𝐴 ∈ V
2		prmg 3697	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴, 𝐵}

Syntax hints:  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by: (None)