Theorem elunii 3794

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 2230	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
2		eleq1 2229	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	anbi12d 465	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶)))
4	3	spcegv 2814	. . 3 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶)))
5	4	anabsi7 571	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶))
6		eluni 3792	. 2 ⊢ (𝐴 ∈ ∪ 𝐶 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶))
7	5, 6	sylibr 133	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∪ cuni 3789

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-uni 3790

This theorem is referenced by:  ssuni  3811  unipw  4195  opeluu  4428  sucunielr  4487  unon  4488  ordunisuc2r  4491  tfrlemibxssdm  6295  tfr1onlemsucaccv  6309  tfr1onlembxssdm  6311  tfrcllemsucaccv  6322  tfrcllembxssdm  6324  tgss2  12719  neipsm  12794  unirnblps  13062  unirnbl  13063  blbas  13073