Theorem elunii 3893

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 2293	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eleq1 2292	. . . . 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 2891	. . 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 3891	. 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 1395  ∃wex 1538   ∈ wcel 2200  ∪ cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-uni 3889

This theorem is referenced by:  ssuni  3910  unipw  4303  opeluu  4541  sucunielr  4602  unon  4603  ordunisuc2r  4606  tfrlemibxssdm  6479  tfr1onlemsucaccv  6493  tfr1onlembxssdm  6495  tfrcllemsucaccv  6506  tfrcllembxssdm  6508  wrdexb  11096  tgss2  14768  neipsm  14843  unirnblps  15111  unirnbl  15112  blbas  15122