Theorem grothprimlem 10788

Description: Lemma for grothprim 10789. 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 4602	. . 3 ⊢ {𝑢, 𝑣} = {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)}
2	1	eleq1i 2852	. 2 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤)
3		clabel 2906	. 2 ⊢ ({ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))
4	2, 3	bitri 277	1 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∨ wo 858  ∀wal 1557  ∃wex 1798   ∈ wcel 2141  {cab 2739  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584

This theorem is referenced by:  grothprim  10789