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Mirrors > Home > MPE Home > Th. List > infxpidm2 | Structured version Visualization version GIF version |
Description: Every infinite well-orderable set is equinumerous to its Cartesian square. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 10505. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
infxpidm2 | β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9896 | . . . . . 6 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
2 | 1 | ensymd 8952 | . . . . 5 β’ (π΄ β dom card β π΄ β (cardβπ΄)) |
3 | xpen 9091 | . . . . 5 β’ ((π΄ β (cardβπ΄) β§ π΄ β (cardβπ΄)) β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) | |
4 | 2, 2, 3 | syl2anc 585 | . . . 4 β’ (π΄ β dom card β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) |
5 | 4 | adantr 482 | . . 3 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) |
6 | cardon 9887 | . . . 4 β’ (cardβπ΄) β On | |
7 | cardom 9929 | . . . . 5 β’ (cardβΟ) = Ο | |
8 | omelon 9589 | . . . . . . . 8 β’ Ο β On | |
9 | onenon 9892 | . . . . . . . 8 β’ (Ο β On β Ο β dom card) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 β’ Ο β dom card |
11 | carddom2 9920 | . . . . . . 7 β’ ((Ο β dom card β§ π΄ β dom card) β ((cardβΟ) β (cardβπ΄) β Ο βΌ π΄)) | |
12 | 10, 11 | mpan 689 | . . . . . 6 β’ (π΄ β dom card β ((cardβΟ) β (cardβπ΄) β Ο βΌ π΄)) |
13 | 12 | biimpar 479 | . . . . 5 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (cardβΟ) β (cardβπ΄)) |
14 | 7, 13 | eqsstrrid 3998 | . . . 4 β’ ((π΄ β dom card β§ Ο βΌ π΄) β Ο β (cardβπ΄)) |
15 | infxpen 9957 | . . . 4 β’ (((cardβπ΄) β On β§ Ο β (cardβπ΄)) β ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) | |
16 | 6, 14, 15 | sylancr 588 | . . 3 β’ ((π΄ β dom card β§ Ο βΌ π΄) β ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) |
17 | entr 8953 | . . 3 β’ (((π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄)) β§ ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) β (π΄ Γ π΄) β (cardβπ΄)) | |
18 | 5, 16, 17 | syl2anc 585 | . 2 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β (cardβπ΄)) |
19 | 1 | adantr 482 | . 2 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (cardβπ΄) β π΄) |
20 | entr 8953 | . 2 β’ (((π΄ Γ π΄) β (cardβπ΄) β§ (cardβπ΄) β π΄) β (π΄ Γ π΄) β π΄) | |
21 | 18, 19, 20 | syl2anc 585 | 1 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β wcel 2107 β wss 3915 class class class wbr 5110 Γ cxp 5636 dom cdm 5638 Oncon0 6322 βcfv 6501 Οcom 7807 β cen 8887 βΌ cdom 8888 cardccrd 9878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-oi 9453 df-card 9882 |
This theorem is referenced by: infpwfien 10005 mappwen 10055 infdjuabs 10149 infxpdom 10154 fin67 10338 infxpidm 10505 ttac 41389 |
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