Theorem prmg 3788

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

Assertion

Ref	Expression
prmg	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem prmg

Step	Hyp	Ref	Expression
1		snmg 3784	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2		orc 717	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
3		velsn 3683	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		vex 2802	. . . . 5 ⊢ 𝑥 ∈ V
5	4	elpr 3687	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
6	2, 3, 5	3imtr4i 201	. . 3 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
7	6	eximi 1646	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {csn 3666  {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prm  3790  opm  4319  onintexmid  4664  subrngin  14171  subrgin  14202  lssincl  14343  wlkvtxiedgg  16042