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Theorem icoreclin 33335
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 33332 . . 3 (𝑦𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})
31icoreelrnab 33332 . . . . . . 7 (𝑥𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})
41isbasisrelowllem1 33333 . . . . . . . . . . . . 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 33334 . . . . . . . . . . . . 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 3838 . . . . . . . . . . . . . . 15 (𝑦𝑥) = (𝑥𝑦)
101isbasisrelowllem2 33334 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑦𝑥) ∈ 𝐼)
119, 10syl5eqelr 2735 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1211ancom1s 864 . . . . . . . . . . . . 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 33333 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑦𝑥) ∈ 𝐼)
159, 14syl5eqelr 2735 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1615ancom1s 864 . . . . . . . . . . . . 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 1078 . . . . . . . . . . . 12 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20 3simpa 1078 . . . . . . . . . . . 12 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21 letric 10175 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎𝑐𝑐𝑎))
22 letric 10175 . . . . . . . . . . . . . . 15 ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏𝑑𝑑𝑏))
2321, 22anim12i 589 . . . . . . . . . . . . . 14 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)))
24 anddi 932 . . . . . . . . . . . . . 14 (((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)) ↔ (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2523, 24sylib 208 . . . . . . . . . . . . 13 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2625an4s 886 . . . . . . . . . . . 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 1286 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼)))
3130rexlimivv 3065 . . . . . . 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 1286 . . . 4 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼)))
3534rexlimivv 3065 . . 3 (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
362, 35sylbi 207 . 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 1054   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  cin 3606   class class class wbr 4685   × cxp 5141  cima 5146  cr 9973   < clt 10112  cle 10113  [,)cico 12215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-ico 12219
This theorem is referenced by:  isbasisrelowl  33336
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