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Theorem fneint 36349
Description: If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
Assertion
Ref Expression
fneint (𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑃

Proof of Theorem fneint
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2w 2825 . . . . 5 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
21elrab 3692 . . . 4 (𝑦 ∈ {𝑥𝐴𝑃𝑥} ↔ (𝑦𝐴𝑃𝑦))
3 fnessex 36347 . . . . . . 7 ((𝐴Fne𝐵𝑦𝐴𝑃𝑦) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
433expb 1121 . . . . . 6 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
5 eleq2w 2825 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
65intminss 4974 . . . . . . . . 9 ((𝑧𝐵𝑃𝑧) → {𝑥𝐵𝑃𝑥} ⊆ 𝑧)
7 sstr 3992 . . . . . . . . 9 (( {𝑥𝐵𝑃𝑥} ⊆ 𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
86, 7sylan 580 . . . . . . . 8 (((𝑧𝐵𝑃𝑧) ∧ 𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
98expl 457 . . . . . . 7 (𝑧𝐵 → ((𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
109rexlimiv 3148 . . . . . 6 (∃𝑧𝐵 (𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
114, 10syl 17 . . . . 5 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1211ex 412 . . . 4 (𝐴Fne𝐵 → ((𝑦𝐴𝑃𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
132, 12biimtrid 242 . . 3 (𝐴Fne𝐵 → (𝑦 ∈ {𝑥𝐴𝑃𝑥} → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
1413ralrimiv 3145 . 2 (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
15 ssint 4964 . 2 ( {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥} ↔ ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1614, 15sylibr 234 1 (𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951   cint 4946   class class class wbr 5143  Fnecfne 36337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topgen 17488  df-fne 36338
This theorem is referenced by: (None)
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