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  4329  pwel  4358  uniopel  4397  iunpw  4626  dmrnssfld  5045  iotaexab  5356  fvssunirng  5710  relfvssunirn  5711  sefvex  5716  riotaexg  6042  pwuninel2  6553  tfrlem9  6590  tfrexlem  6605  sbthlem1  7274  sbthlem2  7275  unirnioo  10377  eltopss  15112  toponss  15129  isbasis3g  15149  baspartn  15153  bastg  15164  tgcl  15167  epttop  15193  difopn  15211  ssntr  15225  isopn3  15228  isopn3i  15238  neiuni  15264  resttopon  15274  restopn2  15286  ssidcn  15313  lmtopcnp  15353  txuni2  15359  hmeoimaf1o  15417  tgioo  15657  bj-elssuniab  16831