Theorem unss12 4138

Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Assertion

Ref	Expression
unss12	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))

Proof of Theorem unss12

Step	Hyp	Ref	Expression
1		unss1 4135	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
2		unss2 4137	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
3	1, 2	sylan9ss 3947	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   ∪ cun 3900   ⊆ wss 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-ss 3919

This theorem is referenced by:  pwssun  5535  fun  6720  f1un  6821  finsschain  9295  trclun  15020  relexpfld  15055  mulgfval  19101  mvdco  19475  dprd2da  20074  dmdprdsplit2lem  20077  lspun  21041  mulsproplem13  28208  mulsproplem14  28209  spanuni  31703  sshhococi  31705  mthmpps  35892  pibt2  37871  mblfinlem3  38118  dochdmj1  41974  mptrcllem  44149  clcnvlem  44159  dfrcl2  44210  relexpss1d  44241  corclrcl  44243  relexp0a  44252  corcltrcl  44275  frege131d  44300