Theorem elssuni 3915

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 3909	. 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 3887

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 3888

This theorem is referenced by:  unissel  3916  ssunieq  3920  pwuni  4275  pwel  4303  uniopel  4342  iunpw  4568  dmrnssfld  4983  iotaexab  5293  fvssunirng  5638  relfvssunirn  5639  sefvex  5644  riotaexg  5951  pwuninel2  6418  tfrlem9  6455  tfrexlem  6470  sbthlem1  7112  sbthlem2  7113  unirnioo  10157  eltopss  14668  toponss  14685  isbasis3g  14705  baspartn  14709  bastg  14720  tgcl  14723  epttop  14749  difopn  14767  ssntr  14781  isopn3  14784  isopn3i  14794  neiuni  14820  resttopon  14830  restopn2  14842  ssidcn  14869  lmtopcnp  14909  txuni2  14915  hmeoimaf1o  14973  tgioo  15213  bj-elssuniab  16085