Theorem prmg 3652

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 3649	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2		orc 702	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
3		velsn 3549	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		vex 2692	. . . . 5 ⊢ 𝑥 ∈ V
5	4	elpr 3553	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
6	2, 3, 5	3imtr4i 200	. . 3 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐴, 𝐵})
7	6	eximi 1580	. 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 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {csn 3532  {cpr 3533

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  prm  3654  opm  4164  onintexmid  4495