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Theorem icoreclin 35828
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 35825 . . 3 (𝑦𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})
31icoreelrnab 35825 . . . . . . 7 (𝑥𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})
41isbasisrelowllem1 35826 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
54ex 413 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
61isbasisrelowllem2 35827 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
76ex 413 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑎𝑐𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
85, 7jaod 857 . . . . . . . . . . 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 4161 . . . . . . . . . . . . . . 15 (𝑦𝑥) = (𝑥𝑦)
101isbasisrelowllem2 35827 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑦𝑥) ∈ 𝐼)
119, 10eqeltrrid 2843 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1211ancom1s 651 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)
1312ex 413 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑏𝑑) → (𝑥𝑦) ∈ 𝐼))
141isbasisrelowllem1 35826 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑦𝑥) ∈ 𝐼)
159, 14eqeltrrid 2843 . . . . . . . . . . . . . 14 ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1615ancom1s 651 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑐𝑎𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)
1716ex 413 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → ((𝑐𝑎𝑑𝑏) → (𝑥𝑦) ∈ 𝐼))
1813, 17jaod 857 . . . . . . . . . . 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 1148 . . . . . . . . . . . 12 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20 3simpa 1148 . . . . . . . . . . . 12 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21 letric 11255 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎𝑐𝑐𝑎))
22 letric 11255 . . . . . . . . . . . . . . 15 ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏𝑑𝑑𝑏))
2321, 22anim12i 613 . . . . . . . . . . . . . 14 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)))
24 anddi 1009 . . . . . . . . . . . . . 14 (((𝑎𝑐𝑐𝑎) ∧ (𝑏𝑑𝑑𝑏)) ↔ (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2523, 24sylib 217 . . . . . . . . . . . . 13 (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2625an4s 658 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎𝑐𝑏𝑑) ∨ (𝑎𝑐𝑑𝑏)) ∨ ((𝑐𝑎𝑏𝑑) ∨ (𝑐𝑎𝑑𝑏))))
2719, 20, 26syl2an 596 . . . . . . . . . . 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 858 . . . . . . . . . 10 (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) → (𝑥𝑦) ∈ 𝐼)
2928ex 413 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
30293expia 1121 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼)))
3130rexlimivv 3196 . . . . . . 7 (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)}) → (𝑥𝑦) ∈ 𝐼))
323, 31sylbi 216 . . . . . 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 1121 . . . 4 ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼)))
3534rexlimivv 3196 . . 3 (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)} → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
362, 35sylbi 216 . 2 (𝑦𝐼 → (𝑥𝐼 → (𝑥𝑦) ∈ 𝐼))
3736impcom 408 1 ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  {crab 3407  cin 3909   class class class wbr 5105   × cxp 5631  cima 5636  cr 11050   < clt 11189  cle 11190  [,)cico 13266
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-ico 13270
This theorem is referenced by:  isbasisrelowl  35829
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