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Theorem grothprimlem 10240
Description: Lemma for grothprim 10241. 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 4567 . . 3 {𝑢, 𝑣} = { ∣ ( = 𝑢 = 𝑣)}
21eleq1i 2906 . 2 ({𝑢, 𝑣} ∈ 𝑤 ↔ { ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤)
3 clabel 2960 . 2 ({ ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
42, 3bitri 278 1 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844  wal 1536  wex 1781  wcel 2115  {cab 2802  {cpr 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-un 3923  df-sn 4549  df-pr 4551
This theorem is referenced by:  grothprim  10241
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