Theorem elunii 3861

Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)

Assertion

Ref	Expression
elunii	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶)

Proof of Theorem elunii

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2270	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eleq1 2269	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶)))
4	3	spcegv 2865	. . 3 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶)))
5	4	anabsi7 581	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶))
6		eluni 3859	. 2 ⊢ (𝐴 ∈ ∪ 𝐶 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶))
7	5, 6	sylibr 134	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∪ cuni 3856

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-uni 3857

This theorem is referenced by:  ssuni  3878  unipw  4269  opeluu  4505  sucunielr  4566  unon  4567  ordunisuc2r  4570  tfrlemibxssdm  6426  tfr1onlemsucaccv  6440  tfr1onlembxssdm  6442  tfrcllemsucaccv  6453  tfrcllembxssdm  6455  wrdexb  11028  tgss2  14626  neipsm  14701  unirnblps  14969  unirnbl  14970  blbas  14980