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Mirrors > Home > MPE Home > Th. List > cardnum | Structured version Visualization version GIF version |
Description: Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in Ο plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.) |
Ref | Expression |
---|---|
cardnum | β’ {π₯ β£ (cardβπ₯) = π₯} = (Ο βͺ ran β΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscard3 10124 | . . . 4 β’ ((cardβπ₯) = π₯ β π₯ β (Ο βͺ ran β΅)) | |
2 | 1 | bicomi 223 | . . 3 β’ (π₯ β (Ο βͺ ran β΅) β (cardβπ₯) = π₯) |
3 | 2 | eqabi 2865 | . 2 β’ (Ο βͺ ran β΅) = {π₯ β£ (cardβπ₯) = π₯} |
4 | 3 | eqcomi 2737 | 1 β’ {π₯ β£ (cardβπ₯) = π₯} = (Ο βͺ ran β΅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 {cab 2705 βͺ cun 3947 ran crn 5683 βcfv 6553 Οcom 7876 cardccrd 9966 β΅cale 9967 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-oi 9541 df-har 9588 df-card 9970 df-aleph 9971 |
This theorem is referenced by: alephprc 10130 |
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