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Mirrors > Home > MPE Home > Th. List > cardalephex | Structured version Visualization version GIF version |
Description: Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.) |
Ref | Expression |
---|---|
cardalephex | β’ (Ο β π΄ β ((cardβπ΄) = π΄ β βπ₯ β On π΄ = (β΅βπ₯))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . 5 β’ ((Ο β π΄ β§ (cardβπ΄) = π΄) β Ο β π΄) | |
2 | cardaleph 10080 | . . . . . . 7 β’ ((Ο β π΄ β§ (cardβπ΄) = π΄) β π΄ = (β΅ββ© {π¦ β On β£ π΄ β (β΅βπ¦)})) | |
3 | 2 | sseq2d 4006 | . . . . . 6 β’ ((Ο β π΄ β§ (cardβπ΄) = π΄) β (Ο β π΄ β Ο β (β΅ββ© {π¦ β On β£ π΄ β (β΅βπ¦)}))) |
4 | alephgeom 10073 | . . . . . 6 β’ (β© {π¦ β On β£ π΄ β (β΅βπ¦)} β On β Ο β (β΅ββ© {π¦ β On β£ π΄ β (β΅βπ¦)})) | |
5 | 3, 4 | bitr4di 289 | . . . . 5 β’ ((Ο β π΄ β§ (cardβπ΄) = π΄) β (Ο β π΄ β β© {π¦ β On β£ π΄ β (β΅βπ¦)} β On)) |
6 | 1, 5 | mpbid 231 | . . . 4 β’ ((Ο β π΄ β§ (cardβπ΄) = π΄) β β© {π¦ β On β£ π΄ β (β΅βπ¦)} β On) |
7 | fveq2 6881 | . . . . 5 β’ (π₯ = β© {π¦ β On β£ π΄ β (β΅βπ¦)} β (β΅βπ₯) = (β΅ββ© {π¦ β On β£ π΄ β (β΅βπ¦)})) | |
8 | 7 | rspceeqv 3625 | . . . 4 β’ ((β© {π¦ β On β£ π΄ β (β΅βπ¦)} β On β§ π΄ = (β΅ββ© {π¦ β On β£ π΄ β (β΅βπ¦)})) β βπ₯ β On π΄ = (β΅βπ₯)) |
9 | 6, 2, 8 | syl2anc 583 | . . 3 β’ ((Ο β π΄ β§ (cardβπ΄) = π΄) β βπ₯ β On π΄ = (β΅βπ₯)) |
10 | 9 | ex 412 | . 2 β’ (Ο β π΄ β ((cardβπ΄) = π΄ β βπ₯ β On π΄ = (β΅βπ₯))) |
11 | alephcard 10061 | . . . 4 β’ (cardβ(β΅βπ₯)) = (β΅βπ₯) | |
12 | fveq2 6881 | . . . 4 β’ (π΄ = (β΅βπ₯) β (cardβπ΄) = (cardβ(β΅βπ₯))) | |
13 | id 22 | . . . 4 β’ (π΄ = (β΅βπ₯) β π΄ = (β΅βπ₯)) | |
14 | 11, 12, 13 | 3eqtr4a 2790 | . . 3 β’ (π΄ = (β΅βπ₯) β (cardβπ΄) = π΄) |
15 | 14 | rexlimivw 3143 | . 2 β’ (βπ₯ β On π΄ = (β΅βπ₯) β (cardβπ΄) = π΄) |
16 | 10, 15 | impbid1 224 | 1 β’ (Ο β π΄ β ((cardβπ΄) = π΄ β βπ₯ β On π΄ = (β΅βπ₯))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwrex 3062 {crab 3424 β wss 3940 β© cint 4940 Oncon0 6354 βcfv 6533 Οcom 7848 cardccrd 9926 β΅cale 9927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-har 9548 df-card 9930 df-aleph 9931 |
This theorem is referenced by: infenaleph 10082 isinfcard 10083 alephfp 10099 alephval3 10101 dfac12k 10138 alephval2 10563 winalim2 10687 minregex 42774 |
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