Theorem elssuni 3916

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 3244	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3910	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2200   ⊆ wss 3197  ∪ cuni 3888

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 2801  df-in 3203  df-ss 3210  df-uni 3889

This theorem is referenced by:  unissel  3917  ssunieq  3921  pwuni  4277  pwel  4305  uniopel  4344  iunpw  4572  dmrnssfld  4990  iotaexab  5300  fvssunirng  5647  relfvssunirn  5648  sefvex  5653  riotaexg  5967  pwuninel2  6439  tfrlem9  6476  tfrexlem  6491  sbthlem1  7140  sbthlem2  7141  unirnioo  10186  eltopss  14704  toponss  14721  isbasis3g  14741  baspartn  14745  bastg  14756  tgcl  14759  epttop  14785  difopn  14803  ssntr  14817  isopn3  14820  isopn3i  14830  neiuni  14856  resttopon  14866  restopn2  14878  ssidcn  14905  lmtopcnp  14945  txuni2  14951  hmeoimaf1o  15009  tgioo  15249  bj-elssuniab  16264