Theorem grothprimlem 10589

Description: Lemma for grothprim 10590. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)

Assertion

Ref	Expression
grothprimlem	⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))

Distinct variable group:   𝑤,𝑣,𝑢,ℎ,𝑔

Proof of Theorem grothprimlem

Step	Hyp	Ref	Expression
1		dfpr2 4580	. . 3 ⊢ {𝑢, 𝑣} = {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)}
2	1	eleq1i 2829	. 2 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤)
3		clabel 2885	. 2 ⊢ ({ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))
4	2, 3	bitri 274	1 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 844  ∀wal 1537  ∃wex 1782   ∈ wcel 2106  {cab 2715  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by:  grothprim  10590