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Mirrors > Home > MPE Home > Th. List > alephinit | Structured version Visualization version Unicode version |
Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
alephinit |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinfcard 8520 |
. . . . 5
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2 | 1 | bicomi 206 |
. . . 4
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3 | 2 | baib 913 |
. . 3
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4 | 3 | adantl 468 |
. 2
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5 | onenon 8380 |
. . . . . . . 8
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6 | 5 | adantr 467 |
. . . . . . 7
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7 | onenon 8380 |
. . . . . . 7
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8 | carddom2 8408 |
. . . . . . 7
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9 | 6, 7, 8 | syl2an 480 |
. . . . . 6
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10 | cardonle 8388 |
. . . . . . . 8
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11 | 10 | adantl 468 |
. . . . . . 7
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12 | sstr 3439 |
. . . . . . . 8
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13 | 12 | expcom 437 |
. . . . . . 7
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14 | 11, 13 | syl 17 |
. . . . . 6
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15 | 9, 14 | sylbird 239 |
. . . . 5
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16 | sseq1 3452 |
. . . . . 6
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17 | 16 | imbi2d 318 |
. . . . 5
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18 | 15, 17 | syl5ibcom 224 |
. . . 4
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19 | 18 | ralrimdva 2805 |
. . 3
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20 | oncardid 8387 |
. . . . . . 7
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21 | ensym 7615 |
. . . . . . 7
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22 | endom 7593 |
. . . . . . 7
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23 | 20, 21, 22 | 3syl 18 |
. . . . . 6
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24 | 23 | adantr 467 |
. . . . 5
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25 | cardon 8375 |
. . . . . 6
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26 | breq2 4405 |
. . . . . . . 8
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27 | sseq2 3453 |
. . . . . . . 8
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28 | 26, 27 | imbi12d 322 |
. . . . . . 7
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29 | 28 | rspcv 3145 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 25, 29 | ax-mp 5 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 24, 30 | syl5com 31 |
. . . 4
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32 | cardonle 8388 |
. . . . . . 7
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33 | 32 | adantr 467 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33 | biantrurd 511 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | eqss 3446 |
. . . . 5
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36 | 34, 35 | syl6bbr 267 |
. . . 4
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37 | 31, 36 | sylibd 218 |
. . 3
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38 | 19, 37 | impbid 194 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 4, 38 | bitrd 257 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-inf2 8143 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-se 4793 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-isom 5590 df-riota 6250 df-om 6690 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-oi 8022 df-har 8070 df-card 8370 df-aleph 8371 |
This theorem is referenced by: (None) |
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