Theorem unssi 3788

Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Hypotheses

Ref	Expression
unssi.1	⊢ 𝐴 ⊆ 𝐶
unssi.2	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
unssi	⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶

Proof of Theorem unssi

Step	Hyp	Ref	Expression
1		unssi.1	. . 3 ⊢ 𝐴 ⊆ 𝐶
2		unssi.2	. . 3 ⊢ 𝐵 ⊆ 𝐶
3	1, 2	pm3.2i 471	. 2 ⊢ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)
4		unss 3787	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
5	3, 4	mpbi 220	1 ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶

Syntax hints:   ∧ wa 384   ∪ cun 3572   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by:  dmrnssfld  5384  tc2  8618  pwxpndom2  9487  ltrelxr  10099  nn0ssre  11296  nn0ssz  11398  dfle2  11980  difreicc  12304  hashxrcl  13148  ramxrcl  15721  strlemor1OLD  15969  strleun  15972  cssincl  20032  leordtval2  21016  lecldbas  21023  comppfsc  21335  aalioulem2  24088  taylfval  24113  axlowdimlem10  25831  shunssji  28228  shsval3i  28247  shjshsi  28351  spanuni  28403  sshhococi  28405  esumcst  30125  hashf2  30146  sxbrsigalem3  30334  signswch  30638  bj-unrab  32922  bj-tagss  32968  poimirlem16  33425  poimirlem19  33428  poimirlem23  33432  poimirlem29  33438  poimirlem31  33440  poimirlem32  33441  mblfinlem3  33448  mblfinlem4  33449  hdmapevec  37127  rtrclex  37924  trclexi  37927  rtrclexi  37928  cnvrcl0  37932  cnvtrcl0  37933  comptiunov2i  37998  cotrclrcl  38034  cncfiooicclem1  40106  fourierdlem62  40385