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Theorem fneint 33254
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 2842 . . . . 5 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
21elrab 3588 . . . 4 (𝑦 ∈ {𝑥𝐴𝑃𝑥} ↔ (𝑦𝐴𝑃𝑦))
3 fnessex 33252 . . . . . . 7 ((𝐴Fne𝐵𝑦𝐴𝑃𝑦) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
433expb 1101 . . . . . 6 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
5 eleq2w 2842 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
65intminss 4771 . . . . . . . . 9 ((𝑧𝐵𝑃𝑧) → {𝑥𝐵𝑃𝑥} ⊆ 𝑧)
7 sstr 3859 . . . . . . . . 9 (( {𝑥𝐵𝑃𝑥} ⊆ 𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
86, 7sylan 572 . . . . . . . 8 (((𝑧𝐵𝑃𝑧) ∧ 𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
98expl 450 . . . . . . 7 (𝑧𝐵 → ((𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
109rexlimiv 3218 . . . . . 6 (∃𝑧𝐵 (𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
114, 10syl 17 . . . . 5 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1211ex 405 . . . 4 (𝐴Fne𝐵 → ((𝑦𝐴𝑃𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
132, 12syl5bi 234 . . 3 (𝐴Fne𝐵 → (𝑦 ∈ {𝑥𝐴𝑃𝑥} → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
1413ralrimiv 3124 . 2 (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
15 ssint 4761 . 2 ( {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥} ↔ ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1614, 15sylibr 226 1 (𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2051  wral 3081  wrex 3082  {crab 3085  wss 3822   cint 4745   class class class wbr 4925  Fnecfne 33242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-int 4746  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-topgen 16571  df-fne 33243
This theorem is referenced by: (None)
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