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Mirrors > Home > MPE Home > Th. List > ordunisuc2 | Structured version Visualization version Unicode version |
Description: An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.) |
Ref | Expression |
---|---|
ordunisuc2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orduninsuc 6702 |
. 2
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2 | ralnex 2846 |
. . 3
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3 | suceloni 6672 |
. . . . . . . . . 10
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4 | eloni 5456 |
. . . . . . . . . 10
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5 | 3, 4 | syl 17 |
. . . . . . . . 9
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6 | ordtri3 5482 |
. . . . . . . . 9
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7 | 5, 6 | sylan2 481 |
. . . . . . . 8
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8 | 7 | con2bid 335 |
. . . . . . 7
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9 | onnbtwn 5537 |
. . . . . . . . . . . . 13
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10 | imnan 428 |
. . . . . . . . . . . . 13
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11 | 9, 10 | sylibr 217 |
. . . . . . . . . . . 12
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12 | 11 | con2d 120 |
. . . . . . . . . . 11
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13 | pm2.21 112 |
. . . . . . . . . . 11
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14 | 12, 13 | syl6 34 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | adantl 472 |
. . . . . . . . 9
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16 | ax-1 6 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | a1i 11 |
. . . . . . . . 9
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18 | 15, 17 | jaod 386 |
. . . . . . . 8
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19 | eloni 5456 |
. . . . . . . . . . . . . 14
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20 | ordtri2or 5541 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | sylan 478 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | ancoms 459 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | orcomd 394 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | adantr 471 |
. . . . . . . . . 10
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25 | ordsssuc2 5534 |
. . . . . . . . . . . . 13
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26 | 25 | biimpd 212 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | adantr 471 |
. . . . . . . . . . 11
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28 | simpr 467 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 27, 28 | orim12d 854 |
. . . . . . . . . 10
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30 | 24, 29 | mpd 15 |
. . . . . . . . 9
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31 | 30 | ex 440 |
. . . . . . . 8
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32 | 18, 31 | impbid 195 |
. . . . . . 7
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33 | 8, 32 | bitr3d 263 |
. . . . . 6
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34 | 33 | pm5.74da 698 |
. . . . 5
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35 | impexp 452 |
. . . . . 6
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36 | simpr 467 |
. . . . . . . 8
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37 | ordelon 5470 |
. . . . . . . . . 10
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38 | 37 | ex 440 |
. . . . . . . . 9
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39 | 38 | ancrd 561 |
. . . . . . . 8
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40 | 36, 39 | impbid2 209 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 40 | imbi1d 323 |
. . . . . 6
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42 | 35, 41 | syl5bbr 267 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 34, 42 | bitrd 261 |
. . . 4
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44 | 43 | ralbidv2 2835 |
. . 3
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45 | 2, 44 | syl5bbr 267 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 1, 45 | bitrd 261 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pr 4656 ax-un 6615 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-br 4419 df-opab 4478 df-tr 4514 df-eprel 4767 df-po 4777 df-so 4778 df-fr 4815 df-we 4817 df-ord 5449 df-on 5450 df-suc 5452 |
This theorem is referenced by: dflim4 6707 limsuc2 35945 |
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