Theorem elunirab 4854

Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Assertion

Ref	Expression
elunirab	⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elunirab

Step	Hyp	Ref	Expression
1		eluniab 4853	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rab 3147	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
3	2	unieqi 4851	. . 3 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
4	3	eleq2i 2904	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
5		df-rex 3144	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)))
6		an12 643	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ (𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
7	6	exbii 1848	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	5, 7	bitri 277	. 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9	1, 4, 8	3bitr4i 305	1 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780   ∈ wcel 2114  {cab 2799  ∃wrex 3139  {crab 3142  ∪ cuni 4838

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 2793

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-in 3943  df-ss 3952  df-uni 4839

This theorem is referenced by:  neiptopuni  21738  cmpcov2  21998  tgcmp  22009  hauscmplem  22014  conncompid  22039  alexsubALT  22659  cvmliftlem15  32545  fnessref  33705  cover2  35004