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Mathbox for Chen-Pang He |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucconi | Structured version Unicode version |
Description: A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
Ref | Expression |
---|---|
onsucconi.1 |
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Ref | Expression |
---|---|
onsucconi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsucconi.1 |
. . 3
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2 | onsuctop 28418 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | elin 3642 |
. . . 4
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5 | elsuci 4888 |
. . . . 5
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6 | 1 | onunisuci 4935 |
. . . . . . 7
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7 | 6 | eqcomi 2465 |
. . . . . 6
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8 | 7 | cldopn 18762 |
. . . . 5
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9 | 1 | onsuci 6554 |
. . . . . . . . . 10
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10 | 9 | oneli 4929 |
. . . . . . . . 9
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11 | elndif 3583 |
. . . . . . . . . . . 12
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12 | on0eln0 4877 |
. . . . . . . . . . . . . 14
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13 | 12 | biimprd 223 |
. . . . . . . . . . . . 13
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14 | 13 | necon1bd 2667 |
. . . . . . . . . . . 12
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15 | ssdif0 3840 |
. . . . . . . . . . . . 13
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16 | 1 | onssneli 4931 |
. . . . . . . . . . . . 13
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17 | 15, 16 | sylbir 213 |
. . . . . . . . . . . 12
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18 | 11, 14, 17 | syl56 34 |
. . . . . . . . . . 11
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19 | 18 | con2d 115 |
. . . . . . . . . 10
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20 | 1 | oneli 4929 |
. . . . . . . . . . . 12
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21 | on0eln0 4877 |
. . . . . . . . . . . . 13
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22 | 21 | biimprd 223 |
. . . . . . . . . . . 12
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23 | 20, 22 | syl 16 |
. . . . . . . . . . 11
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24 | 23 | necon1bd 2667 |
. . . . . . . . . 10
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25 | 19, 24 | sylcom 29 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 10, 25 | syl 16 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | orim1d 835 |
. . . . . . 7
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28 | 27 | impcom 430 |
. . . . . 6
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29 | vex 3075 |
. . . . . . 7
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30 | 29 | elpr 3998 |
. . . . . 6
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31 | 28, 30 | sylibr 212 |
. . . . 5
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32 | 5, 8, 31 | syl2an 477 |
. . . 4
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33 | 4, 32 | sylbi 195 |
. . 3
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34 | 33 | ssriv 3463 |
. 2
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35 | 7 | iscon2 19145 |
. 2
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36 | 3, 34, 35 | mpbir2an 911 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-rab 2805 df-v 3074 df-sbc 3289 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-iota 5484 df-fun 5523 df-fn 5524 df-fv 5529 df-topgen 14496 df-top 18630 df-bases 18632 df-cld 18750 df-con 19143 |
This theorem is referenced by: onsuccon 28423 |
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