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Theorem rintn0 5016
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 4887 . . 3 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋 𝒫 𝐴)
2 ssid 3974 . . . 4 𝒫 𝐴 ⊆ 𝒫 𝐴
3 sspwuni 5008 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
42, 3mpbi 233 . . 3 𝒫 𝐴𝐴
51, 4sstrdi 3964 . 2 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋𝐴)
6 sseqin2 4176 . 2 ( 𝑋𝐴 ↔ (𝐴 𝑋) = 𝑋)
75, 6sylib 221 1 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wne 3014  cin 3918  wss 3919  c0 4275  𝒫 cpw 4521   cuni 4824   cint 4862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4276  df-pw 4523  df-uni 4825  df-int 4863
This theorem is referenced by:  mrerintcl  16864  ismred2  16870
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