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  4276  pwel  4304  uniopel  4343  iunpw  4571  dmrnssfld  4987  iotaexab  5297  fvssunirng  5644  relfvssunirn  5645  sefvex  5650  riotaexg  5964  pwuninel2  6434  tfrlem9  6471  tfrexlem  6486  sbthlem1  7132  sbthlem2  7133  unirnioo  10177  eltopss  14691  toponss  14708  isbasis3g  14728  baspartn  14732  bastg  14743  tgcl  14746  epttop  14772  difopn  14790  ssntr  14804  isopn3  14807  isopn3i  14817  neiuni  14843  resttopon  14853  restopn2  14865  ssidcn  14892  lmtopcnp  14932  txuni2  14938  hmeoimaf1o  14996  tgioo  15236  bj-elssuniab  16179