Theorem unissd 3818

Description: Subclass relationship for subclass union. Deduction form of uniss 3815. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
unissd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
unissd	⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵)

Proof of Theorem unissd

Step	Hyp	Ref	Expression
1		unissd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		uniss 3815	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3121  ∪ cuni 3794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795

This theorem is referenced by:  iotanul  5173  tfrlemibfn  6305  tfrlemiubacc  6307  tfr1onlemssrecs  6316  tfr1onlembfn  6321  tfr1onlemubacc  6323  tfrcllemssrecs  6329  tfrcllembfn  6334  tfrcllemubacc  6336  fiuni  6953  eltg3i  12815  unitg  12821  tgss  12822  ntrss  12878