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Theorem rintn0 4651
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 4534 . . 3 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋 𝒫 𝐴)
2 ssid 3657 . . . 4 𝒫 𝐴 ⊆ 𝒫 𝐴
3 sspwuni 4643 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
42, 3mpbi 220 . . 3 𝒫 𝐴𝐴
51, 4syl6ss 3648 . 2 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋𝐴)
6 sseqin2 3850 . 2 ( 𝑋𝐴 ↔ (𝐴 𝑋) = 𝑋)
75, 6sylib 208 1 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wne 2823  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-uni 4469  df-int 4508
This theorem is referenced by:  mrerintcl  16304  ismred2  16310
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