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Theorem icoreclin 34775
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 𝑧 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasisrelowl.1 . . . 4 𝐼 = ([,) “ (ℝ × ℝ))
21icoreelrnab 34772 . . 3 (𝑦𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})
31icoreelrnab 34772 . . . . . . 7 (𝑥𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})
41isbasisrelowllem1 34773 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
54ex 416 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
61isbasisrelowllem2 34774 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
76ex 416 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
85, 7jaod 856 . . . . . . . . . . 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 4131 . . . . . . . . . . . . . . 15 (𝑦𝑥) = (𝑥𝑦)
101isbasisrelowllem2 34774 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑦𝑥) ∈ 𝐼)
119, 10eqeltrrid 2898 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1211ancom1s 652 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1312ex 416 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
141isbasisrelowllem1 34773 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑦𝑥) ∈ 𝐼)
159, 14eqeltrrid 2898 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1615ancom1s 652 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1716ex 416 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
1813, 17jaod 856 . . . . . . . . . . 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 1145 . . . . . . . . . . . 12 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20 3simpa 1145 . . . . . . . . . . . 12 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21 letric 10733 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎𝑐𝑐𝑎))
22 letric 10733 . . . . . . . . . . . . . . 15 ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏𝑑𝑑𝑏))
2321, 22anim12i 615 . . . . . . . . . . . . . 14 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)))
24 anddi 1008 . . . . . . . . . . . . . 14 (((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)) ↔ (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2523, 24sylib 221 . . . . . . . . . . . . 13 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2625an4s 659 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2719, 20, 26syl2an 598 . . . . . . . . . . 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 857 . . . . . . . . . 10 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (𝑥𝑦) ∈ 𝐼)
2928ex 416 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
30293expia 1118 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼)))
3130rexlimivv 3254 . . . . . . 7 (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
323, 31sylbi 220 . . . . . 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 1118 . . . 4 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼)))
3534rexlimivv 3254 . . 3 (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
362, 35sylbi 220 . 2 (𝑦𝐼 → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
3736impcom 411 1 ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wrex 3110  {crab 3113  cin 3883   class class class wbr 5033   × cxp 5521  cima 5526  cr 10529   < clt 10668  cle 10669  [,)cico 12732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-pre-lttri 10604  ax-pre-lttrn 10605
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-ico 12736
This theorem is referenced by:  isbasisrelowl  34776
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