Theorem ssun1 3166

Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssun1	⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)

Proof of Theorem ssun1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 669	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		elun 3144	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
3	1, 2	sylibr 133	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝐵))
4	3	ssriv 3032	1 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 665   ∈ wcel 1439   ∪ cun 3000   ⊆ wss 3002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015

This theorem is referenced by:  ssun2  3167  ssun3  3168  elun1  3170  inabs  3234  reuun1  3284  un00  3335  undifabs  3365  undifss  3369  snsspr1  3593  snsstp1  3595  snsstp2  3596  prsstp12  3598  exmidundif  4045  sssucid  4253  unexb  4279  dmexg  4712  fvun1  5385  dftpos2  6042  tpostpos2  6046  ac6sfi  6670  caserel  6834  finomni  6859  ressxr  7594  nnssnn0  8739  un0addcl  8769  un0mulcl  8770  nn0ssxnn0  8802  fsumsplit  10864  fsum2d  10892  fsumabs  10922  bdunexb  12115