Theorem eluni 3786

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 2732	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V)
2		elex 2732	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ V)
3	2	adantr 274	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
4	3	exlimiv 1585	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ V)
5		eleq1 2227	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
6	5	anbi1d 461	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
7	6	exbidv 1812	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
8		df-uni 3784	. . 3 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)}
9	7, 8	elab2g 2868	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)))
10	1, 4, 9	pm5.21nii 694	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1342  ∃wex 1479   ∈ wcel 2135  Vcvv 2721  ∪ cuni 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-uni 3784

This theorem is referenced by:  eluni2  3787  elunii  3788  eluniab  3795  uniun  3802  uniin  3803  uniss  3804  unissb  3813  dftr2  4076  unidif0  4140  unipw  4189  uniex2  4408  uniuni  4423  limom  4585  dmuni  4808  fununi  5250  nfvres  5513  elunirn  5728  tfrlem7  6276  tfrexlem  6293  tfrcldm  6322  fiuni  6934  isbasis2g  12584  tgval2  12592  ntreq0  12673  bj-uniex2  13633