Theorem unssad 4193

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4190. Partial converse of unssd 4192. (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 4190	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 234	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simpld 494	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∪ cun 3949   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968

This theorem is referenced by:  naddcllem  8714  ersym  8757  findcard2d  9206  finsschain  9399  r0weon  10052  ackbij1lem16  10274  wunex2  10778  sumsplit  15804  fsumabs  15837  fsumiun  15857  mrieqvlemd  17672  yonedalem1  18317  yonedalem21  18318  yonedalem22  18323  yonffthlem  18327  lsmsp  21085  mplcoe1  22055  mdetunilem9  22626  ordtbas  23200  isufil2  23916  ufileu  23927  filufint  23928  fmfnfm  23966  flimclslem  23992  fclsfnflim  24035  flimfnfcls  24036  imasdsf1olem  24383  limcdif  25911  jensenlem1  27030  jensenlem2  27031  jensen  27032  gsumvsca1  33232  gsumvsca2  33233  qsdrngilem  33522  fldgenfldext  33718  evls1fldgencl  33720  fldextrspunlem1  33725  fldextrspunfld  33726  algextdeglem1  33758  algextdeglem2  33759  algextdeglem3  33760  algextdeglem4  33761  constrextdg2lem  33789  ordtconnlem1  33923  ssmcls  35572  mclsppslem  35588  rngunsnply  43181  mptrcllem  43626  clcnvlem  43636  brtrclfv2  43740  isotone1  44061  dvnprodlem1  45961