Theorem eluniab 4924

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eluniab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4912	. 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
2		nfv 1918	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
3		nfsab1 2718	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4	2, 3	nfan 1903	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})
5		nfv 1918	. . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝜑)
6		eleq2w 2818	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
7		eleq1w 2817	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2714	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	bitrdi 287	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10	6, 9	anbi12d 632	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝑥 ∧ 𝜑)))
11	4, 5, 10	cbvexv1 2339	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))
12	1, 11	bitri 275	1 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 397  ∃wex 1782   ∈ wcel 2107  {cab 2710  ∪ cuni 4909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-uni 4910

This theorem is referenced by:  elunirab  4925  dfiun2gOLD  5035  inuni  5344  elfv  6890  unielxp  8013  frrlem8  8278  frrlem10  8280  wfrlem12OLD  8320  tfrlem9  8385  dfac5lem2  10119  fin23lem30  10337  unisngl  23031  metrest  24033  aannenlem2  25842  fpwrelmapffslem  31957  dfiota3  34895  mptsnunlem  36219  nnoeomeqom  42062