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Mirrors > Home > MPE Home > Th. List > cardid2 | Structured 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 8236 |
. . 3
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2 | ssrab2 3548 |
. . . 4
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3 | fvex 5812 |
. . . . . 6
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4 | 1, 3 | syl6eqelr 2551 |
. . . . 5
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5 | intex 4559 |
. . . . 5
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6 | 4, 5 | sylibr 212 |
. . . 4
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7 | onint 6519 |
. . . 4
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8 | 2, 6, 7 | sylancr 663 |
. . 3
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9 | 1, 8 | eqeltrd 2542 |
. 2
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10 | breq1 4406 |
. . . 4
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11 | 10 | elrab 3224 |
. . 3
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12 | 11 | simprbi 464 |
. 2
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13 | 9, 12 | syl 16 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-fv 5537 df-en 7424 df-card 8223 |
This theorem is referenced by: isnum3 8238 oncardid 8240 cardidm 8243 ficardom 8245 ficardid 8246 cardnn 8247 cardnueq0 8248 carden2a 8250 carden2b 8251 carddomi2 8254 sdomsdomcardi 8255 cardsdomelir 8257 cardsdomel 8258 infxpidm2 8297 dfac8b 8315 numdom 8322 alephnbtwn2 8356 alephsucdom 8363 infenaleph 8375 dfac12r 8429 cardacda 8481 pwsdompw 8487 cff1 8541 cfflb 8542 cflim2 8546 cfss 8548 cfslb 8549 domtriomlem 8725 cardid 8825 cardidg 8826 carden 8829 sdomsdomcard 8838 hargch 8954 gch2 8956 hashkf 12225 |
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