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Theorem xmetec 22049
 Description: The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 7681, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
Hypothesis
Ref Expression
xmeter.1 = (𝐷 “ ℝ)
Assertion
Ref Expression
xmetec ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → [𝑃] = (𝑃(ball‘𝐷)+∞))

Proof of Theorem xmetec
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 xmeter.1 . . . . 5 = (𝐷 “ ℝ)
21xmeterval 22047 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (𝑃 𝑥 ↔ (𝑃𝑋𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
3 3anass 1035 . . . . 5 ((𝑃𝑋𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ) ↔ (𝑃𝑋 ∧ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
43baib 942 . . . 4 (𝑃𝑋 → ((𝑃𝑋𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
52, 4sylan9bb 732 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (𝑃 𝑥 ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
6 vex 3176 . . . . 5 𝑥 ∈ V
76a1i 11 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝑥 ∈ V)
8 elecg 7672 . . . 4 ((𝑥 ∈ V ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃] 𝑃 𝑥))
97, 8sylan 487 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃] 𝑃 𝑥))
10 xblpnf 22011 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (𝑥 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) ∈ ℝ)))
115, 9, 103bitr4d 299 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → (𝑥 ∈ [𝑃] 𝑥 ∈ (𝑃(ball‘𝐷)+∞)))
1211eqrdv 2608 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → [𝑃] = (𝑃(ball‘𝐷)+∞))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  ◡ccnv 5037   “ cima 5041  ‘cfv 5804  (class class class)co 6549  [cec 7627  ℝcr 9814  +∞cpnf 9950  ∞Metcxmt 19552  ballcbl 19554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-ec 7631  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-2 10956  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-psmet 19559  df-xmet 19560  df-bl 19562 This theorem is referenced by:  blssec  22050  blpnfctr  22051
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