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Theorem icoreclin 32829
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 32826 . . 3 (𝑦𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})
31icoreelrnab 32826 . . . . . . 7 (𝑥𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})
41isbasisrelowllem1 32827 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
54ex 450 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
61isbasisrelowllem2 32828 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
76ex 450 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
85, 7jaod 395 . . . . . . . . . . 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 3788 . . . . . . . . . . . . . . 15 (𝑦𝑥) = (𝑥𝑦)
101isbasisrelowllem2 32828 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑦𝑥) ∈ 𝐼)
119, 10syl5eqelr 2709 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1211ancom1s 846 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1312ex 450 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
141isbasisrelowllem1 32827 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑦𝑥) ∈ 𝐼)
159, 14syl5eqelr 2709 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1615ancom1s 846 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1716ex 450 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
1813, 17jaod 395 . . . . . . . . . . 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 1056 . . . . . . . . . . . 12 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20 3simpa 1056 . . . . . . . . . . . 12 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21 letric 10082 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎𝑐𝑐𝑎))
22 letric 10082 . . . . . . . . . . . . . . 15 ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏𝑑𝑑𝑏))
2321, 22anim12i 589 . . . . . . . . . . . . . 14 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)))
24 anddi 913 . . . . . . . . . . . . . 14 (((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)) ↔ (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2523, 24sylib 208 . . . . . . . . . . . . 13 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2625an4s 868 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2719, 20, 26syl2an 494 . . . . . . . . . . 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 396 . . . . . . . . . 10 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (𝑥𝑦) ∈ 𝐼)
2928ex 450 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
30293expia 1264 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼)))
3130rexlimivv 3034 . . . . . . 7 (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
323, 31sylbi 207 . . . . . 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 1264 . . . 4 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼)))
3534rexlimivv 3034 . . 3 (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
362, 35sylbi 207 . 2 (𝑦𝐼 → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
3736impcom 446 1 ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  wrex 2913  {crab 2916  cin 3559   class class class wbr 4618   × cxp 5077  cima 5082  cr 9880   < clt 10019  cle 10020  [,)cico 12116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-pre-lttri 9955  ax-pre-lttrn 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-ico 12120
This theorem is referenced by:  isbasisrelowl  32830
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