Theorem unssbd 4186

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4182. Partial converse of unssd 4184. (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 4182	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 233	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simprd 494	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   ∪ cun 3942   ⊆ wss 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-un 3949  df-ss 3961

This theorem is referenced by:  eldifpw  7771  naddcllem  8697  ertr  8740  finsschain  9390  r0weon  10042  ackbij1lem16  10265  wunfi  10751  wunex2  10768  hashf1lem2  14458  sumsplit  15755  fsum2dlem  15757  fsumabs  15788  fsumrlim  15798  fsumo1  15799  fsumiun  15808  fprod2dlem  15965  mreexexlem3d  17634  yonedalem1  18272  yonedalem21  18273  yonedalem3a  18274  yonedalem4c  18277  yonedalem22  18278  yonedalem3b  18279  yonedainv  18281  yonffthlem  18282  ablfac1eulem  20046  lsmsp  20988  lsppratlem3  21054  mplcoe1  22002  mdetunilem9  22571  filufint  23873  fmfnfmlem4  23910  hausflim  23934  fclsfnflim  23980  fsumcn  24837  itgfsum  25805  jensenlem1  26969  jensenlem2  26970  gsumvsca1  33030  gsumvsca2  33031  qsdrngilem  33311  evls1fldgencl  33491  ordtconnlem1  33658  vhmcls  35309  mclsppslem  35326  rngunsnply  42741  brtrclfv2  43301