Theorem unssbd 4121

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

Proof of Theorem unssbd

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	simprd 496	1 ⊢ (𝜑 → 𝐵 ⊆ 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:  eldifpw  7608  ertr  8500  finsschain  9113  r0weon  9778  ackbij1lem16  10001  wunfi  10487  wunex2  10504  hashf1lem2  14180  sumsplit  15490  fsum2dlem  15492  fsumabs  15523  fsumrlim  15533  fsumo1  15534  fsumiun  15543  fprod2dlem  15700  mreexexlem3d  17365  yonedalem1  18000  yonedalem21  18001  yonedalem3a  18002  yonedalem4c  18005  yonedalem22  18006  yonedalem3b  18007  yonedainv  18009  yonffthlem  18010  ablfac1eulem  19685  lsmsp  20358  lsppratlem3  20421  mplcoe1  21248  mdetunilem9  21779  filufint  23081  fmfnfmlem4  23118  hausflim  23142  fclsfnflim  23188  fsumcn  24043  itgfsum  25001  jensenlem1  26146  jensenlem2  26147  gsumvsca1  31487  gsumvsca2  31488  ordtconnlem1  31882  vhmcls  33536  mclsppslem  33553  naddcllem  33839  rngunsnply  41006  brtrclfv2  41316