Theorem prm 3770

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 3768	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴, 𝐵}

Syntax hints:  ∃wex 1518   ∈ wcel 2180  Vcvv 2779  {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-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-v 2781  df-un 3181  df-sn 3652  df-pr 3653

This theorem is referenced by: (None)