Theorem ssun1 3367

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 717	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		elun 3345	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
3	1, 2	sylibr 134	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝐵))
4	3	ssriv 3228	1 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 713   ∈ wcel 2200   ∪ cun 3195   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  ssun2  3368  ssun3  3369  elun1  3371  inabs  3436  reuun1  3486  un00  3538  undifabs  3568  undifss  3572  snsspr1  3816  snsstp1  3818  snsstp2  3819  prsstp12  3821  exmidundif  4290  sssucid  4506  unexb  4533  dmexg  4988  fvun1  5702  dftpos2  6413  tpostpos2  6417  ac6sfi  7068  caserel  7265  finomni  7318  ressxr  8201  nnssnn0  9383  un0addcl  9413  un0mulcl  9414  nn0ssxnn0  9446  ccatclab  11142  ccatrn  11157  fsumsplit  11933  fsum2d  11961  fsumabs  11991  fprodsplitdc  12122  fprod2d  12149  ennnfonelemss  12996  prdssca  13323  prdsbas  13324  prdsplusg  13325  prdsmulr  13326  lspun  14381  cnfldbas  14539  mpocnfldadd  14540  mpocnfldmul  14542  cnfldcj  14544  cnfldtset  14545  cnfldle  14546  cnfldds  14547  psrplusgg  14657  dvmptfsum  15414  elplyr  15429  lgsdir2lem3  15724  lgsquadlem2  15772  bdunexb  16338