Theorem gbowodd 48822

Description: A weak odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)

Assertion

Ref	Expression
gbowodd	⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Proof of Theorem gbowodd

Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgbow 48819	. 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	simplbi 502	1 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  (class class class)co 7418   + caddc 11196  ℙcprime 16839   Odd codd 48692   GoldbachOddW cgbow 48813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088  df-rab 3414  df-v 3453  df-gbow 48816

This theorem is used by:  gboodd  48824