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Theorem icoreclin 32381
 Description: The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)
Hypothesis
Ref Expression
isbasisrelowl.1 𝐼 = ([,) “ (ℝ × ℝ))
Assertion
Ref Expression
icoreclin ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)
Distinct variable group:   𝑥,𝐼,𝑦

Proof of Theorem icoreclin
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasisrelowl.1 . . . 4 𝐼 = ([,) “ (ℝ × ℝ))
21icoreelrnab 32378 . . 3 (𝑦𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})
31icoreelrnab 32378 . . . . . . 7 (𝑥𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})
41isbasisrelowllem1 32379 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
54ex 449 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
61isbasisrelowllem2 32380 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
76ex 449 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
85, 7jaod 394 . . . . . . . . . . 11 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼))
9 incom 3767 . . . . . . . . . . . . . . 15 (𝑦𝑥) = (𝑥𝑦)
101isbasisrelowllem2 32380 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑦𝑥) ∈ 𝐼)
119, 10syl5eqelr 2693 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1211ancom1s 843 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1312ex 449 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
141isbasisrelowllem1 32379 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑦𝑥) ∈ 𝐼)
159, 14syl5eqelr 2693 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1615ancom1s 843 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1716ex 449 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
1813, 17jaod 394 . . . . . . . . . . 11 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼))
19 3simpa 1051 . . . . . . . . . . . 12 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20 3simpa 1051 . . . . . . . . . . . 12 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21 letric 10016 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎𝑐𝑐𝑎))
22 letric 10016 . . . . . . . . . . . . . . 15 ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏𝑑𝑑𝑏))
2321, 22anim12i 588 . . . . . . . . . . . . . 14 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)))
24 anddi 910 . . . . . . . . . . . . . 14 (((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)) ↔ (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2523, 24sylib 207 . . . . . . . . . . . . 13 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2625an4s 865 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2719, 20, 26syl2an 493 . . . . . . . . . . 11 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
288, 18, 27mpjaod 395 . . . . . . . . . 10 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (𝑥𝑦) ∈ 𝐼)
2928ex 449 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
30293expia 1259 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼)))
3130rexlimivv 3018 . . . . . . 7 (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
323, 31sylbi 206 . . . . . 6 (𝑥𝐼 → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
3332com12 32 . . . . 5 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
34333expia 1259 . . . 4 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼)))
3534rexlimivv 3018 . . 3 (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
362, 35sylbi 206 . 2 (𝑦𝐼 → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
3736impcom 445 1 ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ∩ cin 3539   class class class wbr 4583   × cxp 5036   “ cima 5041  ℝcr 9814   < clt 9953   ≤ cle 9954  [,)cico 12048 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-pre-lttri 9889  ax-pre-lttrn 9890 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-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-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  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-ico 12052 This theorem is referenced by:  isbasisrelowl  32382
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