Theorem prmg 3794

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 3790	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2		orc 719	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
3		velsn 3686	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		vex 2805	. . . . 5 ⊢ 𝑥 ∈ V
5	4	elpr 3690	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
6	2, 3, 5	3imtr4i 201	. . 3 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
7	6	eximi 1648	. 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 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {csn 3669  {cpr 3670

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 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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prm  3796  opm  4326  onintexmid  4671  subrngin  14226  subrgin  14257  lssincl  14398  wlkvtxiedg  16195  wlkvtxiedgg  16196