Theorem elssuni 3961

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 3268	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3955	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 428	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2209   ⊆ wss 3220  ∪ cuni 3933

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3934

This theorem is referenced by:  unissel  3962  ssunieq  3966  pwuni  4327  pwel  4356  uniopel  4395  iunpw  4624  dmrnssfld  5043  iotaexab  5354  fvssunirng  5708  relfvssunirn  5709  sefvex  5714  riotaexg  6036  pwuninel2  6547  tfrlem9  6584  tfrexlem  6599  sbthlem1  7268  sbthlem2  7269  unirnioo  10358  eltopss  15093  toponss  15110  isbasis3g  15130  baspartn  15134  bastg  15145  tgcl  15148  epttop  15174  difopn  15192  ssntr  15206  isopn3  15209  isopn3i  15219  neiuni  15245  resttopon  15255  restopn2  15267  ssidcn  15294  lmtopcnp  15334  txuni2  15340  hmeoimaf1o  15398  tgioo  15638  bj-elssuniab  16802