Theorem unssad 4120

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4117. Partial converse of unssd 4119. (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 4117	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 233	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simpld 495	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∪ cun 3884   ⊆ wss 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3431  df-un 3891  df-in 3893  df-ss 3903

This theorem is referenced by:  ersym  8497  findcard2d  8936  finsschain  9113  r0weon  9778  ackbij1lem16  10001  wunex2  10504  sumsplit  15490  fsumabs  15523  fsumiun  15543  mrieqvlemd  17348  yonedalem1  18000  yonedalem21  18001  yonedalem22  18006  yonffthlem  18010  lsmsp  20358  mplcoe1  21248  mdetunilem9  21779  ordtbas  22353  isufil2  23069  ufileu  23080  filufint  23081  fmfnfm  23119  flimclslem  23145  fclsfnflim  23188  flimfnfcls  23189  imasdsf1olem  23536  limcdif  25050  jensenlem1  26146  jensenlem2  26147  jensen  26148  gsumvsca1  31487  gsumvsca2  31488  ordtconnlem1  31882  ssmcls  33537  mclsppslem  33553  naddcllem  33839  rngunsnply  41006  mptrcllem  41202  clcnvlem  41212  brtrclfv2  41316  isotone1  41639  dvnprodlem1  43468