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Theorem prmg 3480
 Description: A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
prmg (A 𝑉x x {A, B})
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝑉(x)

Proof of Theorem prmg
StepHypRef Expression
1 snmg 3477 . 2 (A 𝑉x x {A})
2 orc 632 . . . 4 (x = A → (x = A x = B))
3 elsn 3382 . . . 4 (x {A} ↔ x = A)
4 vex 2554 . . . . 5 x V
54elpr 3385 . . . 4 (x {A, B} ↔ (x = A x = B))
62, 3, 53imtr4i 190 . . 3 (x {A} → x {A, B})
76eximi 1488 . 2 (x x {A} → x x {A, B})
81, 7syl 14 1 (A 𝑉x x {A, B})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {csn 3367  {cpr 3368 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374 This theorem is referenced by:  prm  3482  opm  3962
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