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Mirrors > Home > MPE Home > Th. List > cardid2 | Structured version Visualization version Unicode version |
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
Ref | Expression |
---|---|
cardid2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardval3 8404 |
. . 3
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2 | ssrab2 3500 |
. . . 4
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3 | fvex 5889 |
. . . . . 6
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4 | 1, 3 | syl6eqelr 2558 |
. . . . 5
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5 | intex 4557 |
. . . . 5
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6 | 4, 5 | sylibr 217 |
. . . 4
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7 | onint 6641 |
. . . 4
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8 | 2, 6, 7 | sylancr 676 |
. . 3
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9 | 1, 8 | eqeltrd 2549 |
. 2
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10 | breq1 4398 |
. . . 4
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11 | 10 | elrab 3184 |
. . 3
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12 | 11 | simprbi 471 |
. 2
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13 | 9, 12 | 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-sep 4518 ax-nul 4527 ax-pr 4639 ax-un 6602 |
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-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 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-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-ord 5433 df-on 5434 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-fv 5597 df-en 7588 df-card 8391 |
This theorem is referenced by: isnum3 8406 oncardid 8408 cardidm 8411 ficardom 8413 ficardid 8414 cardnn 8415 cardnueq0 8416 carden2a 8418 carden2b 8419 carddomi2 8422 sdomsdomcardi 8423 cardsdomelir 8425 cardsdomel 8426 infxpidm2 8466 dfac8b 8480 numdom 8487 alephnbtwn2 8521 alephsucdom 8528 infenaleph 8540 dfac12r 8594 cardacda 8646 pwsdompw 8652 cff1 8706 cfflb 8707 cflim2 8711 cfss 8713 cfslb 8714 domtriomlem 8890 cardid 8990 cardidg 8991 carden 8994 sdomsdomcard 9003 hargch 9116 gch2 9118 hashkf 12555 |
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