Theorem elssuni 3921

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 3247	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3915	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2202   ⊆ wss 3200  ∪ cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894

This theorem is referenced by:  unissel  3922  ssunieq  3926  pwuni  4282  pwel  4310  uniopel  4349  iunpw  4577  dmrnssfld  4995  iotaexab  5305  fvssunirng  5654  relfvssunirn  5655  sefvex  5660  riotaexg  5975  pwuninel2  6448  tfrlem9  6485  tfrexlem  6500  sbthlem1  7156  sbthlem2  7157  unirnioo  10208  eltopss  14739  toponss  14756  isbasis3g  14776  baspartn  14780  bastg  14791  tgcl  14794  epttop  14820  difopn  14838  ssntr  14852  isopn3  14855  isopn3i  14865  neiuni  14891  resttopon  14901  restopn2  14913  ssidcn  14940  lmtopcnp  14980  txuni2  14986  hmeoimaf1o  15044  tgioo  15284  bj-elssuniab  16413