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Mirrors > Home > MPE Home > Th. List > cardinfima | Structured version Visualization version Unicode version |
Description: If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.) |
Ref | Expression |
---|---|
cardinfima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3040 |
. 2
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2 | isinfcard 8541 |
. . . . . . . . . . . . 13
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3 | 2 | bicomi 207 |
. . . . . . . . . . . 12
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4 | 3 | simplbi 467 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | ffn 5739 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | fnfvelrn 6034 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | ex 441 |
. . . . . . . . . . . . . . 15
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | fnima 5704 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | eleq2d 2534 |
. . . . . . . . . . . . . . 15
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10 | 7, 9 | sylibrd 242 |
. . . . . . . . . . . . . 14
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11 | elssuni 4219 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | syl6 33 |
. . . . . . . . . . . . 13
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13 | 12 | imp 436 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 5, 13 | sylan 479 |
. . . . . . . . . . 11
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15 | 4, 14 | sylan9ssr 3432 |
. . . . . . . . . 10
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16 | 15 | anasss 659 |
. . . . . . . . 9
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17 | 16 | a1i 11 |
. . . . . . . 8
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18 | carduniima 8545 |
. . . . . . . . . 10
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19 | iscard3 8542 |
. . . . . . . . . 10
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20 | 18, 19 | syl6ibr 235 |
. . . . . . . . 9
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21 | 20 | adantrd 475 |
. . . . . . . 8
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22 | 17, 21 | jcad 542 |
. . . . . . 7
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23 | isinfcard 8541 |
. . . . . . 7
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24 | 22, 23 | syl6ib 234 |
. . . . . 6
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25 | 24 | exp4d 620 |
. . . . 5
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26 | 25 | imp 436 |
. . . 4
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27 | 26 | rexlimdv 2870 |
. . 3
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28 | 27 | expimpd 614 |
. 2
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29 | 1, 28 | 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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-inf2 8164 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-se 4799 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-isom 5598 df-riota 6270 df-om 6712 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-oi 8043 df-har 8091 df-card 8391 df-aleph 8392 |
This theorem is referenced by: alephfplem4 8556 |
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