Theorem elssuni 3852

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 3190	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3846	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2160   ⊆ wss 3144  ∪ cuni 3824

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825

This theorem is referenced by:  unissel  3853  ssunieq  3857  pwuni  4210  pwel  4236  uniopel  4274  iunpw  4498  dmrnssfld  4908  iotaexab  5214  fvssunirng  5549  relfvssunirn  5550  sefvex  5555  riotaexg  5855  pwuninel2  6306  tfrlem9  6343  tfrexlem  6358  sbthlem1  6985  sbthlem2  6986  unirnioo  10002  eltopss  13961  toponss  13978  isbasis3g  13998  baspartn  14002  bastg  14013  tgcl  14016  epttop  14042  difopn  14060  ssntr  14074  isopn3  14077  isopn3i  14087  neiuni  14113  resttopon  14123  restopn2  14135  ssidcn  14162  lmtopcnp  14202  txuni2  14208  hmeoimaf1o  14266  tgioo  14498  bj-elssuniab  14996