Theorem unssbd 4189

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. 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
unssbd	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Proof of Theorem unssbd

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	simprd 497	1 ⊢ (𝜑 → 𝐵 ⊆ 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:  eldifpw  7755  naddcllem  8675  ertr  8718  finsschain  9359  r0weon  10007  ackbij1lem16  10230  wunfi  10716  wunex2  10733  hashf1lem2  14417  sumsplit  15714  fsum2dlem  15716  fsumabs  15747  fsumrlim  15757  fsumo1  15758  fsumiun  15767  fprod2dlem  15924  mreexexlem3d  17590  yonedalem1  18225  yonedalem21  18226  yonedalem3a  18227  yonedalem4c  18230  yonedalem22  18231  yonedalem3b  18232  yonedainv  18234  yonffthlem  18235  ablfac1eulem  19942  lsmsp  20697  lsppratlem3  20762  mplcoe1  21592  mdetunilem9  22122  filufint  23424  fmfnfmlem4  23461  hausflim  23485  fclsfnflim  23531  fsumcn  24386  itgfsum  25344  jensenlem1  26491  jensenlem2  26492  gsumvsca1  32402  gsumvsca2  32403  qsdrngilem  32639  algextdeglem1  32803  ordtconnlem1  32935  vhmcls  34588  mclsppslem  34605  rngunsnply  41963  brtrclfv2  42526