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Mirrors > Home > MPE Home > Th. List > limuni3 | Structured version Unicode version |
Description: The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.) |
Ref | Expression |
---|---|
limuni3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limeq 4834 |
. . . . . . 7
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2 | 1 | rspcv 3169 |
. . . . . 6
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3 | vex 3075 |
. . . . . . 7
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4 | limelon 4885 |
. . . . . . 7
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5 | 3, 4 | mpan 670 |
. . . . . 6
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6 | 2, 5 | syl6com 35 |
. . . . 5
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7 | 6 | ssrdv 3465 |
. . . 4
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8 | ssorduni 6502 |
. . . 4
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9 | 7, 8 | syl 16 |
. . 3
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10 | 9 | adantl 466 |
. 2
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11 | n0 3749 |
. . . 4
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12 | 0ellim 4884 |
. . . . . . 7
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13 | elunii 4199 |
. . . . . . . 8
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14 | 13 | expcom 435 |
. . . . . . 7
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15 | 12, 14 | syl5 32 |
. . . . . 6
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16 | 2, 15 | syld 44 |
. . . . 5
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17 | 16 | exlimiv 1689 |
. . . 4
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18 | 11, 17 | sylbi 195 |
. . 3
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19 | 18 | imp 429 |
. 2
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20 | eluni2 4198 |
. . . . 5
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21 | 1 | rspccv 3170 |
. . . . . . 7
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22 | limsuc 6565 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 22 | anbi1d 704 |
. . . . . . . . . 10
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24 | elunii 4199 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | syl6bi 228 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | expd 436 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | com3r 79 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 21, 27 | sylcom 29 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28 | rexlimdv 2940 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 20, 29 | syl5bi 217 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | ralrimiv 2825 |
. . 3
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32 | 31 | adantl 466 |
. 2
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33 | dflim4 6564 |
. 2
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34 | 10, 19, 32, 33 | syl3anbrc 1172 |
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-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-tr 4489 df-eprel 4735 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-lim 4827 df-suc 4828 |
This theorem is referenced by: (None) |
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