Theorem eluni2 3897

Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
eluni2	⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eluni2

Step	Hyp	Ref	Expression
1		exancom 1656	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
2		eluni 3896	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
3		df-rex 2516	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
4	1, 2, 3	3bitr4i 212	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  ∪ cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-uni 3894

This theorem is referenced by:  uni0b  3918  intssunim  3950  iuncom4  3977  inuni  4245  ssorduni  4585  unon  4609  cnvuni  4916  chfnrn  5758  zrhval  14637  isbasis3g  14776  eltg2b  14784  tgcl  14794  epttop  14820  txuni2  14986