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Theorem intssuni2 4722
Description: Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssuni2 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)

Proof of Theorem intssuni2
StepHypRef Expression
1 intssuni 4719 . 2 (𝐴 ≠ ∅ → 𝐴 𝐴)
2 uniss 4681 . 2 (𝐴𝐵 𝐴 𝐵)
31, 2sylan9ssr 3841 1 ((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wne 2999  wss 3798  c0 4144   cuni 4658   cint 4697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812  df-nul 4145  df-uni 4659  df-int 4698
This theorem is referenced by:  rintn0  4840  fival  8587  mremre  16617  submre  16618  lssintcl  19323  iundifdifd  29916  iundifdif  29917  bj-ismoored2  33579  bj-ismooredr2  33581  ismrcd1  38098
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