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Theorem fneint 32649
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 2823 . . . . 5 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
21elrab 3504 . . . 4 (𝑦 ∈ {𝑥𝐴𝑃𝑥} ↔ (𝑦𝐴𝑃𝑦))
3 fnessex 32647 . . . . . . 7 ((𝐴Fne𝐵𝑦𝐴𝑃𝑦) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
433expb 1114 . . . . . 6 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
5 eleq2w 2823 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
65intminss 4655 . . . . . . . . 9 ((𝑧𝐵𝑃𝑧) → {𝑥𝐵𝑃𝑥} ⊆ 𝑧)
7 sstr 3752 . . . . . . . . 9 (( {𝑥𝐵𝑃𝑥} ⊆ 𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
86, 7sylan 489 . . . . . . . 8 (((𝑧𝐵𝑃𝑧) ∧ 𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
98expl 649 . . . . . . 7 (𝑧𝐵 → ((𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
109rexlimiv 3165 . . . . . 6 (∃𝑧𝐵 (𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
114, 10syl 17 . . . . 5 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1211ex 449 . . . 4 (𝐴Fne𝐵 → ((𝑦𝐴𝑃𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
132, 12syl5bi 232 . . 3 (𝐴Fne𝐵 → (𝑦 ∈ {𝑥𝐴𝑃𝑥} → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
1413ralrimiv 3103 . 2 (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
15 ssint 4645 . 2 ( {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥} ↔ ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1614, 15sylibr 224 1 (𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  wrex 3051  {crab 3054  wss 3715   cint 4627   class class class wbr 4804  Fnecfne 32637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-topgen 16306  df-fne 32638
This theorem is referenced by: (None)
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