Theorem eluni 3608

Description: Membership in class union. (Contributed by NM, 22-May-1994.)

Assertion

Ref	Expression
eluni	⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eluni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2581	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V)
2		elex 2581	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V)
3	2	adantr 265	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
4	3	exlimiv 1503	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
5		eleq1 2114	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
6	5	anbi1d 446	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
7	6	exbidv 1720	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
8		df-uni 3606	. . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)}
9	7, 8	elab2g 2709	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
10	1, 4, 9	pm5.21nii 628	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ∧ wa 101   ↔ wb 102   = wceq 1257  ∃wex 1395   ∈ wcel 1407  Vcvv 2572  ∪ cuni 3605

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036

This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-uni 3606

This theorem is referenced by:  eluni2  3609  elunii  3610  eluniab  3617  uniun  3624  uniin  3625  uniss  3626  unissb  3635  dftr2  3881  unidif0  3945  unipw  3978  uniex2  4198  uniuni  4208  limom  4361  dmuni  4570  fununi  4992  nfvres  5231  elunirn  5430  tfrlem7  5961  tfrexlem  5976  bj-uniex2  10366