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Theorem grothprimlem 9856
Description: Lemma for grothprim 9857. 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 4332 . . 3 {𝑢, 𝑣} = { ∣ ( = 𝑢 = 𝑣)}
21eleq1i 2840 . 2 ({𝑢, 𝑣} ∈ 𝑤 ↔ { ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤)
3 clabel 2897 . 2 ({ ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
42, 3bitri 264 1 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wo 826  wal 1628  wex 1851  wcel 2144  {cab 2756  {cpr 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-sn 4315  df-pr 4317
This theorem is referenced by:  grothprim  9857
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