Theorem ssun1 1621

Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27.

Assertion

Ref	Expression
ssun1	⊢ A ⊆ (A ∪ B)

Proof of Theorem ssun1

Step	Hyp	Ref	Expression
1		orc 225	. . 3 ⊢ (x ∈ A → (x ∈ A ∨ x ∈ B))
2		elun 1601	. . 3 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
3	1, 2	sylibr 175	. 2 ⊢ (x ∈ A → x ∈ (A ∪ B))
4	3	ssriv 1508	1 ⊢ A ⊆ (A ∪ B)

Syntax hints:   ∨ wo 195   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487

This theorem is referenced by:  ssun2 1622  ssun3 1623  elun1 1625  un00 1728  unexb 1950  sssucid 2300  tfrlem11 2959  mapunen 3397  rankun 3535  cdadom3 3729  nnssnn0 4537  infxpidmlem1 4933  infxpidmlem11 4943  infunabs 4946  infdif 4948  shsumval2 5361  sshhococ 5451

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492

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