Theorem ssun1 3370

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 719	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		elun 3348	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
3	1, 2	sylibr 134	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝐵))
4	3	ssriv 3231	1 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 715   ∈ wcel 2202   ∪ cun 3198   ⊆ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by:  ssun2  3371  ssun3  3372  elun1  3374  inabs  3439  reuun1  3489  un00  3541  undifabs  3571  undifss  3575  snsspr1  3821  snsstp1  3823  snsstp2  3824  prsstp12  3826  exmidundif  4296  sssucid  4512  unexb  4539  dmexg  4996  fvun1  5712  dftpos2  6426  tpostpos2  6430  ac6sfi  7086  caserel  7285  finomni  7338  ressxr  8222  nnssnn0  9404  un0addcl  9434  un0mulcl  9435  nn0ssxnn0  9467  ccatclab  11170  ccatrn  11185  fsumsplit  11967  fsum2d  11995  fsumabs  12025  fprodsplitdc  12156  fprod2d  12183  ennnfonelemss  13030  prdssca  13357  prdsbas  13358  prdsplusg  13359  prdsmulr  13360  lspun  14415  cnfldbas  14573  mpocnfldadd  14574  mpocnfldmul  14576  cnfldcj  14578  cnfldtset  14579  cnfldle  14580  cnfldds  14581  psrplusgg  14691  dvmptfsum  15448  elplyr  15463  lgsdir2lem3  15758  lgsquadlem2  15806  bdunexb  16515