Theorem grothprimlem 9534

Description: Lemma for grothprim 9535. 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 4143	. . 3 ⊢ {𝑢, 𝑣} = {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)}
2	1	eleq1i 2679	. 2 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤)
3		clabel 2736	. 2 ⊢ ({ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))
4	2, 3	bitri 263	1 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))

Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  {cab 2596  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128

This theorem is referenced by:  grothprim  9535