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Theorem rintm 3772
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 3157 . 2 (𝐴 𝑋) = ( 𝑋𝐴)
2 intssuni2m 3667 . . . 4 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋 𝒫 𝐴)
3 ssid 2992 . . . . 5 𝒫 𝐴 ⊆ 𝒫 𝐴
4 sspwuni 3767 . . . . 5 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
53, 4mpbi 137 . . . 4 𝒫 𝐴𝐴
62, 5syl6ss 2985 . . 3 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋𝐴)
7 df-ss 2959 . . 3 ( 𝑋𝐴 ↔ ( 𝑋𝐴) = 𝑋)
86, 7sylib 131 . 2 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑋𝐴) = 𝑋)
91, 8syl5eq 2100 1 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  cin 2944  wss 2945  𝒫 cpw 3387   cuni 3608   cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-uni 3609  df-int 3644
This theorem is referenced by: (None)
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