![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10585. (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 9976 | . . . . . 6 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
2 | 1 | ensymd 9024 | . . . . 5 β’ (π΄ β dom card β π΄ β (cardβπ΄)) |
3 | xpen 9163 | . . . . 5 β’ ((π΄ β (cardβπ΄) β§ π΄ β (cardβπ΄)) β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) | |
4 | 2, 2, 3 | syl2anc 582 | . . . 4 β’ (π΄ β dom card β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) |
5 | 4 | adantr 479 | . . 3 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β ((cardβπ΄) Γ (cardβπ΄))) |
6 | cardon 9967 | . . . 4 β’ (cardβπ΄) β On | |
7 | cardom 10009 | . . . . 5 β’ (cardβΟ) = Ο | |
8 | omelon 9669 | . . . . . . . 8 β’ Ο β On | |
9 | onenon 9972 | . . . . . . . 8 β’ (Ο β On β Ο β dom card) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 β’ Ο β dom card |
11 | carddom2 10000 | . . . . . . 7 β’ ((Ο β dom card β§ π΄ β dom card) β ((cardβΟ) β (cardβπ΄) β Ο βΌ π΄)) | |
12 | 10, 11 | mpan 688 | . . . . . 6 β’ (π΄ β dom card β ((cardβΟ) β (cardβπ΄) β Ο βΌ π΄)) |
13 | 12 | biimpar 476 | . . . . 5 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (cardβΟ) β (cardβπ΄)) |
14 | 7, 13 | eqsstrrid 4022 | . . . 4 β’ ((π΄ β dom card β§ Ο βΌ π΄) β Ο β (cardβπ΄)) |
15 | infxpen 10037 | . . . 4 β’ (((cardβπ΄) β On β§ Ο β (cardβπ΄)) β ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) | |
16 | 6, 14, 15 | sylancr 585 | . . 3 β’ ((π΄ β dom card β§ Ο βΌ π΄) β ((cardβπ΄) Γ (cardβπ΄)) β (cardβπ΄)) |
17 | entr 9025 | . . 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 582 | . 2 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β (cardβπ΄)) |
19 | 1 | adantr 479 | . 2 β’ ((π΄ β dom card β§ Ο βΌ π΄) β (cardβπ΄) β π΄) |
20 | entr 9025 | . 2 β’ (((π΄ Γ π΄) β (cardβπ΄) β§ (cardβπ΄) β π΄) β (π΄ Γ π΄) β π΄) | |
21 | 18, 19, 20 | syl2anc 582 | 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 394 β wcel 2098 β wss 3939 class class class wbr 5143 Γ cxp 5670 dom cdm 5672 Oncon0 6364 βcfv 6543 Οcom 7868 β cen 8959 βΌ cdom 8960 cardccrd 9958 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-oi 9533 df-card 9962 |
This theorem is referenced by: infpwfien 10085 mappwen 10135 infdjuabs 10229 infxpdom 10234 fin67 10418 infxpidm 10585 ttac 42522 |
Copyright terms: Public domain | W3C validator |