Theorem elunirab 3927

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 3926	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rab 2529	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
3	2	unieqi 3924	. . 3 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
4	3	eleq2i 2299	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
5		df-rex 2526	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)))
6		an12 563	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ (𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
7	6	exbii 1654	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	5, 7	bitri 184	. 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9	1, 4, 8	3bitr4i 212	1 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1541   ∈ wcel 2203  {cab 2218  ∃wrex 2521  {crab 2524  ∪ cuni 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2815  df-uni 3915

This theorem is referenced by: (None)