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Mirrors > Home > MPE Home > Th. List > snunioo | Structured version Visualization version Unicode version |
Description: The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
snunioo |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1009 |
. . . 4
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2 | iccid 11671 |
. . . 4
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3 | 1, 2 | syl 17 |
. . 3
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4 | 3 | uneq1d 3555 |
. 2
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5 | simp2 1010 |
. . 3
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6 | xrleid 11439 |
. . . 4
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7 | 1, 6 | syl 17 |
. . 3
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8 | simp3 1011 |
. . 3
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9 | df-icc 11632 |
. . . 4
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10 | df-ioo 11629 |
. . . 4
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11 | xrltnle 9688 |
. . . 4
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12 | df-ico 11631 |
. . . 4
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13 | xrlelttr 11443 |
. . . 4
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14 | xrltle 11438 |
. . . . . 6
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15 | 14 | 3adant1 1027 |
. . . . 5
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16 | 15 | adantld 473 |
. . . 4
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17 | 9, 10, 11, 12, 13, 16 | ixxun 11641 |
. . 3
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18 | 1, 1, 5, 7, 8, 17 | syl32anc 1279 |
. 2
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19 | 4, 18 | eqtr3d 2488 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1673 ax-4 1686 ax-5 1762 ax-6 1809 ax-7 1855 ax-8 1893 ax-9 1900 ax-10 1919 ax-11 1924 ax-12 1937 ax-13 2092 ax-ext 2432 ax-sep 4497 ax-nul 4506 ax-pow 4554 ax-pr 4612 ax-un 6571 ax-cnex 9582 ax-resscn 9583 ax-pre-lttri 9600 ax-pre-lttrn 9601 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 987 df-3an 988 df-tru 1451 df-ex 1668 df-nf 1672 df-sb 1802 df-eu 2304 df-mo 2305 df-clab 2439 df-cleq 2445 df-clel 2448 df-nfc 2582 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3015 df-sbc 3236 df-csb 3332 df-dif 3375 df-un 3377 df-in 3379 df-ss 3386 df-nul 3700 df-if 3850 df-pw 3921 df-sn 3937 df-pr 3939 df-op 3943 df-uni 4169 df-br 4375 df-opab 4434 df-mpt 4435 df-id 4727 df-po 4733 df-so 4734 df-xp 4818 df-rel 4819 df-cnv 4820 df-co 4821 df-dm 4822 df-rn 4823 df-res 4824 df-ima 4825 df-iota 5525 df-fun 5563 df-fn 5564 df-f 5565 df-f1 5566 df-fo 5567 df-f1o 5568 df-fv 5569 df-ov 6279 df-oprab 6280 df-mpt2 6281 df-er 7350 df-en 7557 df-dom 7558 df-sdom 7559 df-pnf 9664 df-mnf 9665 df-xr 9666 df-ltxr 9667 df-le 9668 df-ioo 11629 df-ico 11631 df-icc 11632 |
This theorem is referenced by: prunioo 11752 ioojoin 11754 icombl1 22528 ioombl 22530 tan2h 31939 mbfposadd 31990 itg2addnclem2 31996 ftc1anclem5 32023 iocunico 36097 limciccioolb 37743 fourierdlem32 38059 fourierdlem93 38120 |
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