Theorem unssad 4188

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4185. Partial converse of unssd 4187. (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 4185	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 233	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simpld 496	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   ∪ cun 3947   ⊆ wss 3949

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966

This theorem is referenced by:  naddcllem  8675  ersym  8715  findcard2d  9166  finsschain  9359  r0weon  10007  ackbij1lem16  10230  wunex2  10733  sumsplit  15714  fsumabs  15747  fsumiun  15767  mrieqvlemd  17573  yonedalem1  18225  yonedalem21  18226  yonedalem22  18231  yonffthlem  18235  lsmsp  20697  mplcoe1  21592  mdetunilem9  22122  ordtbas  22696  isufil2  23412  ufileu  23423  filufint  23424  fmfnfm  23462  flimclslem  23488  fclsfnflim  23531  flimfnfcls  23532  imasdsf1olem  23879  limcdif  25393  jensenlem1  26491  jensenlem2  26492  jensen  26493  gsumvsca1  32402  gsumvsca2  32403  qsdrngilem  32639  algextdeglem1  32803  ordtconnlem1  32935  ssmcls  34589  mclsppslem  34605  rngunsnply  41963  mptrcllem  42412  clcnvlem  42422  brtrclfv2  42526  isotone1  42847  dvnprodlem1  44710