Theorem 3unrab 32485

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 4264	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ ((𝜑 ∨ 𝜓) ∨ 𝜒)}
2		unrab 4264	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)}
3	2	uneq1i 4113	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = ({𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} ∪ {𝑥 ∈ 𝐴 ∣ 𝜒})
4		df-3or 1087	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
5	4	rabbii 3401	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)} = {𝑥 ∈ 𝐴 ∣ ((𝜑 ∨ 𝜓) ∨ 𝜒)}
6	1, 3, 5	3eqtr4i 2766	1 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)}

Syntax hints:   ∨ wo 847   ∨ w3o 1085   = wceq 1541  {crab 3396   ∪ cun 3896

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-un 3903

This theorem is referenced by:  constrfin  33780