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Theorem rintn0 5114
Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
rintn0 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)

Proof of Theorem rintn0
StepHypRef Expression
1 intssuni2 4978 . . 3 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋 𝒫 𝐴)
2 ssid 4018 . . . 4 𝒫 𝐴 ⊆ 𝒫 𝐴
3 sspwuni 5105 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
42, 3mpbi 230 . . 3 𝒫 𝐴𝐴
51, 4sstrdi 4008 . 2 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋𝐴)
6 sseqin2 4231 . 2 ( 𝑋𝐴 ↔ (𝐴 𝑋) = 𝑋)
75, 6sylib 218 1 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wne 2938  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952
This theorem is referenced by:  mrerintcl  17642  ismred2  17648
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