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Theorem rintm 3735
 Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
rintm ((𝑋 ⊆ 𝒫 A x x 𝑋) → (A 𝑋) = 𝑋)
Distinct variable group:   x,𝑋
Allowed substitution hint:   A(x)

Proof of Theorem rintm
StepHypRef Expression
1 incom 3123 . 2 (A 𝑋) = ( 𝑋A)
2 intssuni2m 3630 . . . 4 ((𝑋 ⊆ 𝒫 A x x 𝑋) → 𝑋 𝒫 A)
3 ssid 2958 . . . . 5 𝒫 A ⊆ 𝒫 A
4 sspwuni 3730 . . . . 5 (𝒫 A ⊆ 𝒫 A 𝒫 AA)
53, 4mpbi 133 . . . 4 𝒫 AA
62, 5syl6ss 2951 . . 3 ((𝑋 ⊆ 𝒫 A x x 𝑋) → 𝑋A)
7 df-ss 2925 . . 3 ( 𝑋A ↔ ( 𝑋A) = 𝑋)
86, 7sylib 127 . 2 ((𝑋 ⊆ 𝒫 A x x 𝑋) → ( 𝑋A) = 𝑋)
91, 8syl5eq 2081 1 ((𝑋 ⊆ 𝒫 A x x 𝑋) → (A 𝑋) = 𝑋)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911  𝒫 cpw 3351  ∪ cuni 3571  ∩ cint 3606 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572  df-int 3607 This theorem is referenced by: (None)
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