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Mirrors > Home > MPE Home > Th. List > domtriord | Structured version Visualization version Unicode version |
Description: Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.) |
Ref | Expression |
---|---|
domtriord |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbth 7692 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | expcom 437 |
. . . 4
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3 | 2 | a1i 11 |
. . 3
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4 | iman 426 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | brsdom 7592 |
. . . 4
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6 | 4, 5 | xchbinxr 313 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 3, 6 | syl6ib 230 |
. 2
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8 | onelss 5465 |
. . . . . . . . . 10
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9 | ssdomg 7615 |
. . . . . . . . . 10
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10 | 8, 9 | syld 45 |
. . . . . . . . 9
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11 | 10 | adantl 468 |
. . . . . . . 8
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12 | 11 | con3d 139 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | ontri1 5457 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | ancoms 455 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 12, 14 | sylibrd 238 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | ssdomg 7615 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | adantr 467 |
. . . . . 6
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18 | 15, 17 | syld 45 |
. . . . 5
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19 | ensym 7618 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | endom 7596 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | syl 17 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | con3i 141 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | jcad 536 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24, 5 | syl6ibr 231 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | con1d 128 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 7, 26 | impbid 194 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-ord 5426 df-on 5427 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 |
This theorem is referenced by: sdomel 7719 cardsdomel 8408 alephord 8506 alephsucdom 8510 alephdom2 8518 |
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