Theorem unssad 3790

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3787. Partial converse of unssd 3789. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
unssad.1	⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)

Assertion

Ref	Expression
unssad	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem unssad

Step	Hyp	Ref	Expression
1		unssad.1	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)
2		unss 3787	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 224	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simpld 475	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 384   ∪ cun 3572   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by:  ersym  7754  findcard2d  8202  finsschain  8273  r0weon  8835  ackbij1lem16  9057  wunex2  9560  sumsplit  14499  fsumabs  14533  fsumiun  14553  mrieqvlemd  16289  yonedalem1  16912  yonedalem21  16913  yonedalem22  16918  yonffthlem  16922  lsmsp  19086  mplcoe1  19465  mdetunilem9  20426  ordtbas  20996  isufil2  21712  ufileu  21723  filufint  21724  fmfnfm  21762  flimclslem  21788  fclsfnflim  21831  flimfnfcls  21832  imasdsf1olem  22178  mbfeqalem  23409  limcdif  23640  jensenlem1  24713  jensenlem2  24714  jensen  24715  gsumvsca1  29782  gsumvsca2  29783  ordtconnlem1  29970  ssmcls  31464  mclsppslem  31480  rngunsnply  37743  mptrcllem  37920  clcnvlem  37930  brtrclfv2  38019  isotone1  38346  dvnprodlem1  40161