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Theorem rintm 3991
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 3339 . 2 (𝐴 𝑋) = ( 𝑋𝐴)
2 intssuni2m 3880 . . . 4 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋 𝒫 𝐴)
3 ssid 3187 . . . . 5 𝒫 𝐴 ⊆ 𝒫 𝐴
4 sspwuni 3983 . . . . 5 (𝒫 𝐴 ⊆ 𝒫 𝐴 𝒫 𝐴𝐴)
53, 4mpbi 145 . . . 4 𝒫 𝐴𝐴
62, 5sstrdi 3179 . . 3 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → 𝑋𝐴)
7 df-ss 3154 . . 3 ( 𝑋𝐴 ↔ ( 𝑋𝐴) = 𝑋)
86, 7sylib 122 . 2 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → ( 𝑋𝐴) = 𝑋)
91, 8eqtrid 2232 1 ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wex 1502  wcel 2158  cin 3140  wss 3141  𝒫 cpw 3587   cuni 3821   cint 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589  df-uni 3822  df-int 3857
This theorem is referenced by: (None)
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