Theorem elssuni 3947

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 3262	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3941	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2205   ⊆ wss 3214  ∪ cuni 3919

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-uni 3920

This theorem is referenced by:  unissel  3948  ssunieq  3952  pwuni  4310  pwel  4339  uniopel  4378  iunpw  4606  dmrnssfld  5025  iotaexab  5336  fvssunirng  5690  relfvssunirn  5691  sefvex  5696  riotaexg  6015  pwuninel2  6526  tfrlem9  6563  tfrexlem  6578  sbthlem1  7240  sbthlem2  7241  unirnioo  10325  eltopss  14986  toponss  15003  isbasis3g  15023  baspartn  15027  bastg  15038  tgcl  15041  epttop  15067  difopn  15085  ssntr  15099  isopn3  15102  isopn3i  15112  neiuni  15138  resttopon  15148  restopn2  15160  ssidcn  15187  lmtopcnp  15227  txuni2  15233  hmeoimaf1o  15291  tgioo  15531  bj-elssuniab  16675