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Mirrors > Home > MPE Home > Th. List > iscard | Structured version Visualization version Unicode version |
Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Ref | Expression |
---|---|
iscard |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 8403 |
. . 3
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2 | eleq1 2527 |
. . 3
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3 | 1, 2 | mpbii 216 |
. 2
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4 | cardonle 8416 |
. . . 4
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5 | eqss 3458 |
. . . . 5
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6 | 5 | baibr 920 |
. . . 4
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7 | 4, 6 | syl 17 |
. . 3
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8 | onelon 5466 |
. . . . . 6
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9 | onenon 8408 |
. . . . . . 7
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10 | 9 | adantr 471 |
. . . . . 6
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11 | cardsdomel 8433 |
. . . . . 6
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12 | 8, 10, 11 | syl2anc 671 |
. . . . 5
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13 | 12 | ralbidva 2835 |
. . . 4
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14 | dfss3 3433 |
. . . 4
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15 | 13, 14 | syl6rbbr 272 |
. . 3
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16 | 7, 15 | bitr3d 263 |
. 2
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17 | 3, 16 | biadan2 652 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-ord 5444 df-on 5445 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-er 7388 df-en 7595 df-dom 7596 df-sdom 7597 df-card 8398 |
This theorem is referenced by: cardprclem 8438 cardmin2 8457 infxpenlem 8469 alephsuc2 8536 cardmin 9014 alephreg 9032 pwcfsdom 9033 winalim2 9146 gchina 9149 inar1 9225 r1tskina 9232 gruina 9268 |
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