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Theorem mreincl 16865
Description: Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mreincl ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Proof of Theorem mreincl
StepHypRef Expression
1 intprg 4908 . . 3 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1124 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} = (𝐴𝐵))
3 simp1 1130 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → 𝐶 ∈ (Moore‘𝑋))
4 prssi 4753 . . . 4 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
543adant1 1124 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
6 prnzg 4712 . . . 4 (𝐴𝐶 → {𝐴, 𝐵} ≠ ∅)
763ad2ant2 1128 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ≠ ∅)
8 mreintcl 16861 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝐴, 𝐵} ⊆ 𝐶 ∧ {𝐴, 𝐵} ≠ ∅) → {𝐴, 𝐵} ∈ 𝐶)
93, 5, 7, 8syl3anc 1365 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝐶)
102, 9eqeltrrd 2919 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wne 3021  cin 3939  wss 3940  c0 4295  {cpr 4566   cint 4874  cfv 6354  Moorecmre 16848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-mre 16852
This theorem is referenced by:  submacs  17986  subgacs  18258  nsgacs  18259  lsmmod  18737  subrgacs  19515  sdrgacs  19516  lssacs  19675  mreclatdemoBAD  21639
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