Theorem unssad 3381

Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3378. Partial converse of unssd 3380. (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 3378	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
3	1, 2	sylibr 134	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
4	3	simpld 112	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∪ cun 3195   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  ersym  6700  findcard2d  7061  findcard2sd  7062  diffifi  7064  sumsplitdc  11951  fsumabs  11984  fsumiun  11996