Theorem unssbd 4194

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

Proof of Theorem unssbd

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	simprd 495	1 ⊢ (𝜑 → 𝐵 ⊆ 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:  eldifpw  7788  naddcllem  8714  ertr  8760  finsschain  9399  r0weon  10052  ackbij1lem16  10274  wunfi  10761  wunex2  10778  hashf1lem2  14495  sumsplit  15804  fsum2dlem  15806  fsumabs  15837  fsumrlim  15847  fsumo1  15848  fsumiun  15857  fprod2dlem  16016  mreexexlem3d  17689  yonedalem1  18317  yonedalem21  18318  yonedalem3a  18319  yonedalem4c  18322  yonedalem22  18323  yonedalem3b  18324  yonedainv  18326  yonffthlem  18327  ablfac1eulem  20092  lsmsp  21085  lsppratlem3  21151  mplcoe1  22055  mdetunilem9  22626  filufint  23928  fmfnfmlem4  23965  hausflim  23989  fclsfnflim  24035  fsumcn  24894  itgfsum  25862  jensenlem1  27030  jensenlem2  27031  gsumvsca1  33232  gsumvsca2  33233  qsdrngilem  33522  evls1fldgencl  33720  fldextrspunlem1  33725  constrextdg2lem  33789  ordtconnlem1  33923  vhmcls  35571  mclsppslem  35588  rngunsnply  43181  brtrclfv2  43740