Theorem elssuni 3919

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 3245	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3913	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2200   ⊆ wss 3198  ∪ cuni 3891

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 2802  df-in 3204  df-ss 3211  df-uni 3892

This theorem is referenced by:  unissel  3920  ssunieq  3924  pwuni  4280  pwel  4308  uniopel  4347  iunpw  4575  dmrnssfld  4993  iotaexab  5303  fvssunirng  5650  relfvssunirn  5651  sefvex  5656  riotaexg  5970  pwuninel2  6443  tfrlem9  6480  tfrexlem  6495  sbthlem1  7150  sbthlem2  7151  unirnioo  10201  eltopss  14726  toponss  14743  isbasis3g  14763  baspartn  14767  bastg  14778  tgcl  14781  epttop  14807  difopn  14825  ssntr  14839  isopn3  14842  isopn3i  14852  neiuni  14878  resttopon  14888  restopn2  14900  ssidcn  14927  lmtopcnp  14967  txuni2  14973  hmeoimaf1o  15031  tgioo  15271  bj-elssuniab  16337