Theorem elssuni 3881

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 3215	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3875	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 424	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2177   ⊆ wss 3168  ∪ cuni 3853

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3174  df-ss 3181  df-uni 3854

This theorem is referenced by:  unissel  3882  ssunieq  3886  pwuni  4241  pwel  4267  uniopel  4306  iunpw  4532  dmrnssfld  4947  iotaexab  5256  fvssunirng  5601  relfvssunirn  5602  sefvex  5607  riotaexg  5913  pwuninel2  6378  tfrlem9  6415  tfrexlem  6430  sbthlem1  7071  sbthlem2  7072  unirnioo  10108  eltopss  14531  toponss  14548  isbasis3g  14568  baspartn  14572  bastg  14583  tgcl  14586  epttop  14612  difopn  14630  ssntr  14644  isopn3  14647  isopn3i  14657  neiuni  14683  resttopon  14693  restopn2  14705  ssidcn  14732  lmtopcnp  14772  txuni2  14778  hmeoimaf1o  14836  tgioo  15076  bj-elssuniab  15841