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Theorem grothprimlem 10243
Description: Lemma for grothprim 10244. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
Assertion
Ref Expression
grothprimlem ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
Distinct variable group:   𝑤,𝑣,𝑢,,𝑔

Proof of Theorem grothprimlem
StepHypRef Expression
1 dfpr2 4576 . . 3 {𝑢, 𝑣} = { ∣ ( = 𝑢 = 𝑣)}
21eleq1i 2900 . 2 ({𝑢, 𝑣} ∈ 𝑤 ↔ { ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤)
3 clabel 2956 . 2 ({ ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
42, 3bitri 276 1 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841  wal 1526  wex 1771  wcel 2105  {cab 2796  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  grothprim  10244
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