Theorem grothprimlem 10257

Description: Lemma for grothprim 10258. 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 4588	. . 3 ⊢ {𝑢, 𝑣} = {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)}
2	1	eleq1i 2905	. 2 ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ {ℎ ∣ (ℎ = 𝑢 ∨ ℎ = 𝑣)} ∈ 𝑤)
3		clabel 2961	. 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 398   ∨ wo 843  ∀wal 1535  ∃wex 1780   ∈ wcel 2114  {cab 2801  {cpr 4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-un 3943  df-sn 4570  df-pr 4572

This theorem is referenced by:  grothprim  10258