Theorem 3unrab 32598

Description: Union of three restricted class abstractions. (Contributed by Thierry Arnoux, 6-Jul-2025.)

Assertion

Ref	Expression
3unrab	⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)}

Proof of Theorem 3unrab

Step	Hyp	Ref	Expression
1		unrab 4250	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ ((𝜑 ∨ 𝜓) ∨ 𝜒)}
2		unrab 4250	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)}
3	2	uneq1i 4101	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = ({𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} ∪ {𝑥 ∈ 𝐴 ∣ 𝜒})
4		df-3or 1093	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
5	4	rabbii 3397	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)} = {𝑥 ∈ 𝐴 ∣ ((𝜑 ∨ 𝜓) ∨ 𝜒)}
6	1, 3, 5	3eqtr4i 2773	1 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)}

Syntax hints:   ∨ wo 853   ∨ w3o 1091   = wceq 1547  {crab 3392   ∪ cun 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-un 3895

This theorem is referenced by:  constrfin  33937