Theorem ssun1 3280

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 702	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		elun 3258	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
3	1, 2	sylibr 133	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝐵))
4	3	ssriv 3141	1 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 698   ∈ wcel 2135   ∪ cun 3109   ⊆ wss 3111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124

This theorem is referenced by:  ssun2  3281  ssun3  3282  elun1  3284  inabs  3349  reuun1  3399  un00  3450  undifabs  3480  undifss  3484  snsspr1  3715  snsstp1  3717  snsstp2  3718  prsstp12  3720  exmidundif  4179  sssucid  4387  unexb  4414  dmexg  4862  fvun1  5546  dftpos2  6220  tpostpos2  6224  ac6sfi  6855  caserel  7043  finomni  7095  ressxr  7933  nnssnn0  9108  un0addcl  9138  un0mulcl  9139  nn0ssxnn0  9171  fsumsplit  11334  fsum2d  11362  fsumabs  11392  fprodsplitdc  11523  fprod2d  11550  ennnfonelemss  12280  bdunexb  13637