Theorem elssuni 3687

Description: An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
elssuni	⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Proof of Theorem elssuni

Step	Hyp	Ref	Expression
1		ssid 3045	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3681	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 416	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1439   ⊆ wss 3000  ∪ cuni 3659

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660

This theorem is referenced by:  unissel  3688  ssunieq  3692  pwuni  4033  pwel  4054  uniopel  4092  iunpw  4315  dmrnssfld  4709  fvssunirng  5333  relfvssunirn  5334  sefvex  5339  riotaexg  5626  pwuninel2  6061  tfrlem9  6098  tfrexlem  6113  sbthlem1  6720  sbthlem2  6721  unirnioo  9452  eltopss  11769  toponss  11785  isbasis3g  11805  baspartn  11809  bastg  11822  tgcl  11825  epttop  11851  difopn  11869  ssntr  11883  isopn3  11886  isopn3i  11896  bj-elssuniab  11964