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Mirrors > Home > MPE Home > Th. List > ordunifi | Structured version Visualization version Unicode version |
Description: The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.) |
Ref | Expression |
---|---|
ordunifi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epweon 6610 |
. . . . . 6
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2 | weso 4825 |
. . . . . 6
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3 | 1, 2 | ax-mp 5 |
. . . . 5
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4 | soss 4773 |
. . . . 5
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5 | 3, 4 | mpi 20 |
. . . 4
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6 | fimax2g 7817 |
. . . 4
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7 | 5, 6 | syl3an1 1301 |
. . 3
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8 | ssel2 3427 |
. . . . . . . . 9
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9 | 8 | adantlr 721 |
. . . . . . . 8
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10 | ssel2 3427 |
. . . . . . . . 9
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11 | 10 | adantr 467 |
. . . . . . . 8
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12 | ontri1 5457 |
. . . . . . . . 9
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13 | epel 4748 |
. . . . . . . . . 10
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14 | 13 | notbii 298 |
. . . . . . . . 9
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15 | 12, 14 | syl6rbbr 268 |
. . . . . . . 8
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16 | 9, 11, 15 | syl2anc 667 |
. . . . . . 7
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17 | 16 | ralbidva 2824 |
. . . . . 6
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18 | unissb 4229 |
. . . . . 6
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19 | 17, 18 | syl6bbr 267 |
. . . . 5
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20 | 19 | rexbidva 2898 |
. . . 4
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21 | 20 | 3ad2ant1 1029 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 7, 21 | mpbid 214 |
. 2
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23 | elssuni 4227 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | eqss 3447 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | eleq1 2517 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 25 | biimpcd 228 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 24, 26 | syl5bir 222 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 23, 27 | mpand 681 |
. . 3
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29 | 28 | rexlimiv 2873 |
. 2
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30 | 22, 29 | syl 17 |
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 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-tp 3973 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-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-om 6693 df-1o 7182 df-er 7363 df-en 7570 df-fin 7573 |
This theorem is referenced by: nnunifi 7822 oemapvali 8189 ttukeylem6 8944 limsucncmpi 31105 |
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