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Theorem fneint 36330
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 2822 . . . . 5 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
21elrab 3694 . . . 4 (𝑦 ∈ {𝑥𝐴𝑃𝑥} ↔ (𝑦𝐴𝑃𝑦))
3 fnessex 36328 . . . . . . 7 ((𝐴Fne𝐵𝑦𝐴𝑃𝑦) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
433expb 1119 . . . . . 6 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
5 eleq2w 2822 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
65intminss 4978 . . . . . . . . 9 ((𝑧𝐵𝑃𝑧) → {𝑥𝐵𝑃𝑥} ⊆ 𝑧)
7 sstr 4003 . . . . . . . . 9 (( {𝑥𝐵𝑃𝑥} ⊆ 𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
86, 7sylan 580 . . . . . . . 8 (((𝑧𝐵𝑃𝑧) ∧ 𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
98expl 457 . . . . . . 7 (𝑧𝐵 → ((𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
109rexlimiv 3145 . . . . . 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 3142 . 2 (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
15 ssint 4968 . 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 2105  wral 3058  wrex 3067  {crab 3432  wss 3962   cint 4950   class class class wbr 5147  Fnecfne 36318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-topgen 17489  df-fne 36319
This theorem is referenced by: (None)
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