Theorem elssuni 3963

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 3268	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssuni 3957	. 2 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ ∪ 𝐵)
3	1, 2	mpan 428	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209   ⊆ wss 3220  ∪ cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936

This theorem is used by:  unissel  3964  ssunieq  3968  pwuni  4331  pwel  4360  uniopel  4399  iunpw  4628  dmrnssfld  5047  iotaexab  5359  fvssunirng  5713  relfvssunirn  5714  sefvex  5719  riotaexg  6045  pwuninel2  6558  tfrlem9  6595  tfrexlem  6610  sbthlem1  7281  sbthlem2  7282  unirnioo  10391  eltopss  15252  toponss  15269  isbasis3g  15289  baspartn  15293  bastg  15304  tgcl  15307  epttop  15333  difopn  15351  ssntr  15365  isopn3  15368  isopn3i  15378  neiuni  15404  resttopon  15414  restopn2  15426  ssidcn  15453  lmtopcnp  15493  txuni2  15499  hmeoimaf1o  15557  tgioo  15797  bj-elssuniab  17041