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Theorem elixx1 12400
Description: Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
ixx.1 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
Assertion
Ref Expression
elixx1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐶,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem elixx1
StepHypRef Expression
1 ixx.1 . . . 4 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
21ixxval 12399 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
32eleq2d 2869 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)}))
4 breq2 4846 . . . . 5 (𝑧 = 𝐶 → (𝐴𝑅𝑧𝐴𝑅𝐶))
5 breq1 4845 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑆𝐵𝐶𝑆𝐵))
64, 5anbi12d 618 . . . 4 (𝑧 = 𝐶 → ((𝐴𝑅𝑧𝑧𝑆𝐵) ↔ (𝐴𝑅𝐶𝐶𝑆𝐵)))
76elrab 3557 . . 3 (𝐶 ∈ {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ↔ (𝐶 ∈ ℝ* ∧ (𝐴𝑅𝐶𝐶𝑆𝐵)))
8 3anass 1109 . . 3 ((𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵) ↔ (𝐶 ∈ ℝ* ∧ (𝐴𝑅𝐶𝐶𝑆𝐵)))
97, 8bitr4i 269 . 2 (𝐶 ∈ {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)} ↔ (𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵))
103, 9syl6bb 278 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {crab 3098   class class class wbr 4842  (class class class)co 6872  cmpt2 6874  *cxr 10356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pr 5094  ax-un 7177  ax-cnex 10275  ax-resscn 10276
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ral 3099  df-rex 3100  df-rab 3103  df-v 3391  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4843  df-opab 4905  df-id 5217  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-xr 10361
This theorem is referenced by:  elixx3g  12404  ixxssixx  12405  ixxdisj  12406  ixxun  12407  ixxss1  12409  ixxss2  12410  ixxss12  12411  ixxub  12412  ixxlb  12413  elioo1  12431  elioc1  12433  elico1  12434  elicc1  12435
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