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Theorem rintn0 5109
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 4973 . . 3 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋 𝒫 𝐴)
2 ssid 4006 . . . 4 𝒫 𝐴 ⊆ 𝒫 𝐴
3 sspwuni 5100 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
42, 3mpbi 230 . . 3 𝒫 𝐴𝐴
51, 4sstrdi 3996 . 2 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋𝐴)
6 sseqin2 4223 . 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 1540  wne 2940  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-int 4947
This theorem is referenced by:  mrerintcl  17640  ismred2  17646
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