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Theorem rintn0 4751
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 4634 . . 3 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋 𝒫 𝐴)
2 ssid 3771 . . . 4 𝒫 𝐴 ⊆ 𝒫 𝐴
3 sspwuni 4743 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
42, 3mpbi 220 . . 3 𝒫 𝐴𝐴
51, 4syl6ss 3762 . 2 ((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → 𝑋𝐴)
6 sseqin2 3966 . 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 382   = wceq 1630  wne 2942  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295   cuni 4572   cint 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-v 3351  df-dif 3724  df-in 3728  df-ss 3735  df-nul 4062  df-pw 4297  df-uni 4573  df-int 4610
This theorem is referenced by:  mrerintcl  16464  ismred2  16470
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