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Theorem rintm 3741
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 3126 . 2 (𝐴 𝑋) = ( 𝑋𝐴)
2 intssuni2m 3636 . . . 4 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋 𝒫 𝐴)
3 ssid 2961 . . . . 5 𝒫 𝐴 ⊆ 𝒫 𝐴
4 sspwuni 3736 . . . . 5 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
53, 4mpbi 133 . . . 4 𝒫 𝐴𝐴
62, 5syl6ss 2954 . . 3 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋𝐴)
7 df-ss 2928 . . 3 ( 𝑋𝐴 ↔ ( 𝑋𝐴) = 𝑋)
86, 7sylib 127 . 2 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑋𝐴) = 𝑋)
91, 8syl5eq 2084 1 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wex 1381  wcel 1393  cin 2913  wss 2914  𝒫 cpw 3356   cuni 3577   cint 3612
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-in 2921  df-ss 2928  df-pw 3358  df-uni 3578  df-int 3613
This theorem is referenced by: (None)
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