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Theorem rintm 3869
Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
rintm ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Distinct variable group:   𝑥,𝑋
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem rintm
StepHypRef Expression
1 incom 3232 . 2 (𝐴 𝑋) = ( 𝑋𝐴)
2 intssuni2m 3759 . . . 4 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋 𝒫 𝐴)
3 ssid 3081 . . . . 5 𝒫 𝐴 ⊆ 𝒫 𝐴
4 sspwuni 3861 . . . . 5 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
53, 4mpbi 144 . . . 4 𝒫 𝐴𝐴
62, 5syl6ss 3073 . . 3 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋𝐴)
7 df-ss 3048 . . 3 ( 𝑋𝐴 ↔ ( 𝑋𝐴) = 𝑋)
86, 7sylib 121 . 2 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑋𝐴) = 𝑋)
91, 8syl5eq 2157 1 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wex 1449  wcel 1461  cin 3034  wss 3035  𝒫 cpw 3474   cuni 3700   cint 3735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476  df-uni 3701  df-int 3736
This theorem is referenced by: (None)
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