Theorem elunirab 3713

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 3712	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
2		df-rab 2397	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
3	2	unieqi 3710	. . 3 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
4	3	eleq2i 2179	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
5		df-rex 2394	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)))
6		an12 533	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ (𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
7	6	exbii 1565	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝐴 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	5, 7	bitri 183	. 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9	1, 4, 8	3bitr4i 211	1 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1449   ∈ wcel 1461  {cab 2099  ∃wrex 2389  {crab 2392  ∪ cuni 3700

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-rab 2397  df-v 2657  df-uni 3701

This theorem is referenced by: (None)