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  6557  tfrlem9  6594  tfrexlem  6609  sbthlem1  7279  sbthlem2  7280  unirnioo  10392  eltopss  15235  toponss  15252  isbasis3g  15272  baspartn  15276  bastg  15287  tgcl  15290  epttop  15316  difopn  15334  ssntr  15348  isopn3  15351  isopn3i  15361  neiuni  15387  resttopon  15397  restopn2  15409  ssidcn  15436  lmtopcnp  15476  txuni2  15482  hmeoimaf1o  15540  tgioo  15780  bj-elssuniab  17024